From 3c6e929d4ea00b980c3a74655c9d2cecff0de930 Mon Sep 17 00:00:00 2001 From: Jayesh Tanna Date: Fri, 9 Jan 2026 12:47:30 +0530 Subject: [PATCH 1/4] Adding rft cookbook --- .gitignore | 6 + Demos/RFT_Math_Reasoning/README.md | 192 ++ Demos/RFT_Math_Reasoning/requirements.txt | 13 + .../rft_math_reasoning.ipynb | 769 +++++ Demos/RFT_Math_Reasoning/training_rft.jsonl | 2661 +++++++++++++++++ Demos/RFT_Math_Reasoning/validation_rft.jsonl | 565 ++++ 6 files changed, 4206 insertions(+) create mode 100644 .gitignore create mode 100644 Demos/RFT_Math_Reasoning/README.md create mode 100644 Demos/RFT_Math_Reasoning/requirements.txt create mode 100644 Demos/RFT_Math_Reasoning/rft_math_reasoning.ipynb create mode 100644 Demos/RFT_Math_Reasoning/training_rft.jsonl create mode 100644 Demos/RFT_Math_Reasoning/validation_rft.jsonl diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..526765c --- /dev/null +++ b/.gitignore @@ -0,0 +1,6 @@ +**/__pycache__ +__pycache__ +env*/ +.env +.vs/ +.vscode/ \ No newline at end of file diff --git a/Demos/RFT_Math_Reasoning/README.md b/Demos/RFT_Math_Reasoning/README.md new file mode 100644 index 0000000..3c2c8e8 --- /dev/null +++ b/Demos/RFT_Math_Reasoning/README.md @@ -0,0 +1,192 @@ +# Reinforcement Fine-Tuning with OpenR1-Math-220k Dataset + +This cookbook demonstrates how to fine-tune language models using **Reinforcement Fine-Tuning (RFT)** with the OpenR1-Math-220k dataset on Azure AI. This dataset contains 220,000 advanced mathematical reasoning problems with verified step-by-step solutions, making it ideal for teaching models complex mathematical problem-solving. + +## Overview + +Reinforcement Fine-Tuning (RFT) is a powerful technique for training language models on tasks where: +- Multiple valid solution paths exist +- Correctness can be verified automatically +- Step-by-step reasoning is crucial +- The task requires structured, multi-hop thinking + +This cookbook uses the **OpenR1-Math-220k dataset**, which contains advanced mathematics problems from college-level and competition mathematics. For RFT, we use only the problem statements and final answers extracted from this dataset. The problems cover diverse mathematical domains including algebra, calculus, geometry, and number theory. + +## Dataset Information + +**Source**: [OpenR1-Math-220k on Kaggle](https://www.kaggle.com/datasets/alejopaullier/openr1-math-220k) | [Hugging Face](https://huggingface.co/datasets/open-r1/OpenR1-Math-220k) + +**Dataset Statistics**: +- **Full dataset**: 220,000 mathematical reasoning problems (7 parquet files, 1.45 GB total) +- **Source parquet**: train-00000-of-00007.parquet (13,391 problems with 27,614 verified reasoning traces) +- **Training set**: 2,661 examples (one entry per problem) +- **Validation set**: 565 examples (one entry per problem) + +> **Important**: +> - The RFT dataset contains processed data extracted from verified parquet files +> - Each training example contains **prompts only** (system + user) and a ground truth answer + + +**What the Data Contains**: +The RFT dataset consists of advanced mathematical problems with ground truth answers. Each training example includes: +- **System prompt**: Instructions for mathematical problem-solving behavior +- **Problem statement**: Complex mathematics question requiring multi-step reasoning +- **Ground truth answer**: Final answer extracted from verified solutions, used by the grader for scoring + +**Note**: The RFT format contains only prompts (system + user) and answers. The model learns to generate step-by-step reasoning through reinforcement learning, not by mimicking provided reasoning traces. + +**Problem Domains Covered**: +- Algebra and polynomial factorization +- Calculus and optimization +- Probability theory and expected values +- Geometry and trigonometry +- Number theory and combinatorics +- Linear algebra and matrix operations +- Complex numbers and abstract mathematics + +**Task Complexity**: +- Advanced multi-step mathematical reasoning +- Problems require abstract algebraic manipulation +- Multi-hop logical reasoning (typically 5-15 reasoning steps) +- Advanced mathematical concepts beyond basic arithmetic +- The model must generate detailed reasoning chains (averaging 8k tokens, complex problems up to 16k tokens) + +**Why RFT is Perfect for This Task**: +- **Multiple valid solution paths**: Many ways to solve each problem +- **Verifiable correctness**: Mathematical answers can be automatically checked +- **Exploration-based learning**: Model discovers effective reasoning strategies through reinforcement +- **Quality over memorization**: Model learns reasoning patterns, not just answers + +## What You'll Learn + +This cookbook teaches you how to: + +1. Understand the RFT dataset format (prompts + ground truth answers) +2. Set up your Azure AI environment for RFT +3. Create a grading function to evaluate mathematical reasoning quality +4. Configure and launch an RFT fine-tuning job +5. Monitor training progress and model performance +6. Deploy and test your fine-tuned mathematical reasoning model + +**Note**: The RFT-formatted dataset files (training_rft.jsonl and validation_rft.jsonl) are already provided. If you want to prepare your own dataset from the original Kaggle files, you'll need to extract problems and answers into the RFT format shown in the Dataset Format section. + +## Prerequisites + +- Azure subscription with Microsoft Foundry project (requires **Azure AI User** role) +- Python 3.9 or higher +- Familiarity with Jupyter notebooks +- Kaggle account to download the OpenR1-Math-220k dataset +- Understanding of basic mathematical concepts + +## Supported Models + +RFT in Azure AI Foundry supports the following models: + +- **o4-mini** +- **gpt-5 (PrPr)** + + +> **Note**: Model availability may vary by region. Check the [Azure OpenAI model availability](https://learn.microsoft.com/azure/ai-services/openai/concepts/models) page for current regional support. + +## Files in This Cookbook + +- **README.md**: This file - comprehensive documentation +- **requirements.txt**: Python dependencies required for the cookbook +- **rft_math_reasoning.ipynb**: Step-by-step notebook implementation +- **training_rft.jsonl**: Training dataset (2,661 examples in RFT format) +- **validation_rft.jsonl**: Validation dataset (565 examples in RFT format) + +## Quick Start + + +### 1. Install Dependencies + +```powershell +pip install -r requirements.txt +``` + +### 2. Prepare the Dataset + +You can use the training and validation dataset as is from this directory or you can prepare your own dataset. + +### 3. Set Up Environment Variables + +Create a `.env` file in the root of this directory with your Azure credentials: + +```env +# Required for RFT Fine-Tuning +AZURE_AI_PROJECT_ENDPOINT= +AZURE_SUBSCRIPTION_ID= +AZURE_RESOURCE_GROUP= +AZURE_AOAI_ACCOUNT= +MODEL_NAME= +``` + +### 4. Run the Notebook + +Open `rft_math_reasoning.ipynb` and follow the step-by-step instructions. + +## Dataset Format + +The RFT format for mathematical reasoning follows this structure: + +```json +{ + "messages": [ + { + "role": "user", + "content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nCall a scalene triangle K disguisable if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle integral if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares." + } + ], + "answer": "9" +} +``` + +Each training example contains: +- **messages**: Array with exactly 1 message: + - **user message**: Combined system prompt and mathematical problem statement. The system instructions are included at the beginning of the user content, followed by the actual problem. +- **answer**: Ground truth final answer (used by the grader for verification) + +**Important**: Unlike Supervised Fine-Tuning (SFT), RFT format does NOT include an assistant message. The model learns to generate solutions through reinforcement signals from the grader, which compares the model's output against the ground truth answer field. + +## Training Configuration + +Recommended hyperparameters for RFT with mathematical reasoning: + +- **Model**: o4-mini +- **Epochs**: 2 (prevents overfitting while allowing reinforcement learning) +- **Batch Size**: 1 (limited by memory constraints with long sequence lengths) +- **Learning Rate Multiplier**: 1.0 (standard learning rate for RFT) + +These can be adjusted based on your computational resources and specific requirements. + +## Grading Mathematical Solutions + +The grading function for RFT is a deterministic Python function that evaluates: + +1. **Answer Extraction**: Extracts the final answer from the model's `\boxed{}` notation +2. **Answer Normalization**: Handles different formats (comma-separated, space-separated, LaTeX expressions) +3. **Answer Correctness**: Compares normalized model answer against ground truth + +The grader returns: +- **1.0**: Model answer matches ground truth (after normalization) +- **0.0**: Model answer does not match or is missing/malformed + +This binary reward signal guides the model to learn effective reasoning strategies that produce correct final answers. + +## Expected Outcomes + +After fine-tuning with RFT on OpenR1-Math-220k, your model should: + +- Generate detailed step-by-step mathematical reasoning chains with clear intermediate steps +- Solve college-level and competition mathematics problems across diverse domains +- Produce properly formatted answers with `\boxed{}` notation for grader verification +- Demonstrate significantly improved performance on multi-hop mathematical reasoning compared to the base model + + +## Additional Resources + +- [Azure OpenAI Fine-Tuning Documentation](https://learn.microsoft.com/azure/ai-services/openai/how-to/fine-tuning) +- [OpenR1-Math-220k dataset](https://www.kaggle.com/datasets/alejopaullier/openr1-math-220k) + +--- diff --git a/Demos/RFT_Math_Reasoning/requirements.txt b/Demos/RFT_Math_Reasoning/requirements.txt new file mode 100644 index 0000000..bd9c840 --- /dev/null +++ b/Demos/RFT_Math_Reasoning/requirements.txt @@ -0,0 +1,13 @@ +# Azure AI SDK for fine-tuning +azure-ai-projects>=2.0.0b1 + +# OpenAI SDK +openai + +# Azure Cognitive Services for model deployment +azure-identity +azure-mgmt-cognitiveservices + +# Core libraries +python-dotenv + diff --git a/Demos/RFT_Math_Reasoning/rft_math_reasoning.ipynb b/Demos/RFT_Math_Reasoning/rft_math_reasoning.ipynb new file mode 100644 index 0000000..c9eb68a --- /dev/null +++ b/Demos/RFT_Math_Reasoning/rft_math_reasoning.ipynb @@ -0,0 +1,769 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Reinforcement Fine-Tuning (RFT) with OpenR1-Math-220k Dataset\n", + "\n", + "This notebook demonstrates how to fine-tune language models using **Reinforcement Fine-Tuning (RFT)** with the OpenR1-Math-220k dataset - a collection of 220,000 advanced mathematical reasoning problems with verified step-by-step solutions.\n", + "\n", + "## What You'll Learn\n", + "1. Understand reinforcement fine-tuning (RFT) and how it differs from supervised fine-tuning (SFT)\n", + "2. Define a grader/reward function for mathematical reasoning\n", + "3. Prepare and upload mathematical reasoning datasets\n", + "4. Create and configure an RFT job using the OpenAI method\n", + "5. Monitor training progress and evaluate model performance\n", + "6. Deploy and test your RFT fine-tuned model\n", + "\n", + "**Note**: Execute each cell in sequence." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 1. Setup and Installation\n", + "\n", + "Install all required packages from requirements.txt" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Collecting azure-ai-projects>=2.0.0b1 (from -r requirements.txt (line 2))\n", + " Using cached azure_ai_projects-2.0.0b3-py3-none-any.whl.metadata (68 kB)\n", + "Collecting openai (from -r requirements.txt (line 5))\n", + " Using cached openai-2.14.0-py3-none-any.whl.metadata (29 kB)\n", + "Collecting azure-identity (from -r requirements.txt (line 8))\n", + " Using cached azure_identity-1.25.1-py3-none-any.whl.metadata (88 kB)\n", + "Collecting azure-mgmt-cognitiveservices (from -r requirements.txt (line 9))\n", + " Using cached azure_mgmt_cognitiveservices-14.1.0-py3-none-any.whl.metadata (32 kB)\n", + "Collecting python-dotenv (from -r requirements.txt (line 12))\n", + " Using cached python_dotenv-1.2.1-py3-none-any.whl.metadata (25 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cryptography-46.0.3 distro-1.9.0 h11-0.16.0 httpcore-1.0.9 httpx-0.28.1 idna-3.11 isodate-0.7.2 jiter-0.12.0 msal-1.34.0 msal-extensions-1.3.1 msrest-0.7.1 oauthlib-3.3.1 openai-2.14.0 pycparser-2.23 pydantic-2.12.5 pydantic-core-2.41.5 python-dotenv-1.2.1 requests-2.32.5 requests-oauthlib-2.0.0 sniffio-1.3.1 tqdm-4.67.1 typing-inspection-0.4.2 urllib3-2.6.3\n", + "Note: you may need to restart the kernel to use updated packages.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\n", + "[notice] A new release of pip is available: 24.0 -> 25.3\n", + "[notice] To update, run: python.exe -m pip install --upgrade pip\n" + ] + } + ], + "source": [ + "pip install -r requirements.txt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 2. Import Libraries" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "All libraries imported successfully\n" + ] + } + ], + "source": [ + "import os\n", + "from dotenv import load_dotenv\n", + "from azure.ai.projects import AIProjectClient\n", + "from azure.identity import DefaultAzureCredential\n", + "\n", + "print(\"All libraries imported successfully\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 3. Configure Azure Environment\n", + "\n", + "Set your Azure AI Project endpoint, model name and other environment variables. We're using **o4-mini** in this example for RFT.\n", + "\n", + "Create a .env file with:\n", + "\n", + "```\n", + "AZURE_AI_PROJECT_ENDPOINT=\n", + "AZURE_SUBSCRIPTION_ID=\n", + "AZURE_RESOURCE_GROUP=\n", + "AZURE_AOAI_ACCOUNT=\n", + "MODEL_NAME=o4-mini\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "load_dotenv()\n", + "\n", + "endpoint = os.environ.get(\"AZURE_AI_PROJECT_ENDPOINT\")\n", + "model_name = os.environ.get(\"MODEL_NAME\", \"o4-mini\")\n", + "\n", + "# Define RFT dataset file paths\n", + "training_file_path = \"training_rft.jsonl\"\n", + "validation_file_path = \"validation_rft.jsonl\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 4. Connect to Azure AI Project\n", + "\n", + "Connect to Azure AI Project using Azure credential authentication. This initializes the project client and OpenAI client needed for fine-tuning workflows.\n", + "\n", + "**Important**: Ensure you have the **Azure AI User** role assigned to your account for the Azure AI Project resource." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Connected to Azure AI Project\n" + ] + } + ], + "source": [ + "credential = DefaultAzureCredential()\n", + "project_client = AIProjectClient(endpoint=endpoint, credential=credential)\n", + "openai_client = project_client.get_openai_client()\n", + "\n", + "print(\"Connected to Azure AI Project\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 5. Define Mathematical Grader for RFT\n", + "\n", + "Reinforcement Fine-Tuning (RFT) requires a grader function to evaluate model outputs. Unlike SFT which learns from examples, RFT learns from reward signals.\n", + "\n", + "For mathematical reasoning, we use a **Python-based grader** that deterministically verifies the correctness of the final answer by:\n", + "1. Extracting the answer from the model's response using `\\\\boxed{}` notation\n", + "2. Comparing it against the reference answer from the training data\n", + "3. Returning a score of 1.0 for correct answers and 0.0 for incorrect ones\n", + "\n", + "This approach is more appropriate for math problems than LLM-based scoring, as mathematical correctness is objective." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Python RFT grader configured successfully\n" + ] + } + ], + "source": [ + "# Python-based grader for Azure OpenAI RFT\n", + "# Compares final answers using \\\\boxed{} notation only\n", + "\n", + "grading_function = \"\"\"import re\n", + "\n", + "def normalize(ans: str):\n", + " try:\n", + " if not isinstance(ans, str):\n", + " return []\n", + " parts = re.split(r\"[,\\s]+\", ans.strip())\n", + " return sorted(p for p in parts if p)\n", + " except Exception:\n", + " return []\n", + "\n", + "\n", + "def extract_model_answer(text: str):\n", + " try:\n", + " if not text or not isinstance(text, str):\n", + " return \"\"\n", + " \n", + " pattern = r\"\\\\\\\\boxed\\\\{\"\n", + " matches = list(re.finditer(pattern, text))\n", + " if not matches:\n", + " return \"\"\n", + " \n", + " last_match = matches[-1]\n", + " start = last_match.end()\n", + " \n", + " brace_count = 1\n", + " i = start\n", + " while i < len(text) and brace_count > 0:\n", + " if text[i] == '{':\n", + " brace_count += 1\n", + " elif text[i] == '}':\n", + " brace_count -= 1\n", + " i += 1\n", + " \n", + " if brace_count == 0:\n", + " return text[start:i-1].strip()\n", + " \n", + " return \"\"\n", + " except Exception:\n", + " return \"\"\n", + "\n", + "\n", + "def grade(sample, item):\n", + " try:\n", + " # Get model output - handle both dict and object access\n", + " if isinstance(sample, dict):\n", + " output_text = sample.get(\"output_text\", \"\") or sample.get(\"output_json\", \"\")\n", + " else:\n", + " output_text = getattr(sample, \"output_text\", \"\") or getattr(sample, \"output_json\", \"\")\n", + " \n", + " # Get reference answer\n", + " if isinstance(item, dict):\n", + " ref_raw = item.get(\"answer\", \"\")\n", + " else:\n", + " ref_raw = getattr(item, \"answer\", \"\")\n", + " \n", + " # Handle None or empty values\n", + " if not output_text:\n", + " return 0.0\n", + " if not ref_raw:\n", + " return 0.0\n", + " \n", + " # Convert output_json to string if it's a dict/object\n", + " if isinstance(output_text, dict):\n", + " output_text = str(output_text)\n", + " \n", + " pred_raw = extract_model_answer(str(output_text))\n", + " \n", + " if not pred_raw:\n", + " return 0.0\n", + "\n", + " pred = normalize(pred_raw)\n", + " ref = normalize(str(ref_raw))\n", + "\n", + " return 1.0 if pred == ref else 0.0\n", + " except Exception:\n", + " # Always return 0.0 on any error to prevent job failure\n", + " return 0.0\n", + "\"\"\"\n", + "\n", + "grader = {\n", + " \"type\": \"python\",\n", + " \"source\": grading_function\n", + "}\n", + "\n", + "print(\"Python RFT grader configured successfully\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 5.1. Test Grader Function (Optional)\n", + "\n", + "Test the grader locally with a sample from the training data to verify it works correctly before submitting the RFT job." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "PASS | Simple numeric answer - CORRECT\n", + "PASS | Simple numeric answer - WRONG\n", + "PASS | Missing \\boxed{} - should FAIL even if answer is correct\n", + "PASS | Multiple \\boxed{} - uses LAST one\n", + "PASS | Comma-separated values - order doesn't matter\n", + "PASS | Space-separated values - normalized\n", + "PASS | LaTeX expression\n", + "PASS | Empty \\boxed{}\n", + "ALL TESTS PASSED\n" + ] + } + ], + "source": [ + "exec(grading_function, globals())\n", + "\n", + "test_cases = [\n", + " {\n", + " \"name\": \"Simple numeric answer - CORRECT\",\n", + " \"sample\": {\"output_text\": \"The final answer is \\\\boxed{42}\"},\n", + " \"item\": {\"answer\": \"42\"},\n", + " \"expected\": 1.0\n", + " },\n", + " {\n", + " \"name\": \"Simple numeric answer - WRONG\",\n", + " \"sample\": {\"output_text\": \"The final answer is \\\\boxed{99}\"},\n", + " \"item\": {\"answer\": \"42\"},\n", + " \"expected\": 0.0\n", + " },\n", + " {\n", + " \"name\": \"Missing \\\\boxed{} - should FAIL even if answer is correct\",\n", + " \"sample\": {\"output_text\": \"The answer is 42\"},\n", + " \"item\": {\"answer\": \"42\"},\n", + " \"expected\": 0.0\n", + " },\n", + " {\n", + " \"name\": \"Multiple \\\\boxed{} - uses LAST one\",\n", + " \"sample\": {\"output_text\": \"First: \\\\boxed{wrong}, Final: \\\\boxed{42}\"},\n", + " \"item\": {\"answer\": \"42\"},\n", + " \"expected\": 1.0\n", + " },\n", + " {\n", + " \"name\": \"Comma-separated values - order doesn't matter\",\n", + " \"sample\": {\"output_text\": \"Answer: \\\\boxed{3, 2, 1}\"},\n", + " \"item\": {\"answer\": \"1, 2, 3\"},\n", + " \"expected\": 1.0\n", + " },\n", + " {\n", + " \"name\": \"Space-separated values - normalized\",\n", + " \"sample\": {\"output_text\": \"Answer: \\\\boxed{9 6 4}\"},\n", + " \"item\": {\"answer\": \"4, 6, 9\"},\n", + " \"expected\": 1.0\n", + " },\n", + " {\n", + " \"name\": \"LaTeX expression\",\n", + " \"sample\": {\"output_text\": \"Solution: \\\\boxed{\\\\sqrt{2}}\"},\n", + " \"item\": {\"answer\": \"\\\\sqrt{2}\"},\n", + " \"expected\": 1.0\n", + " },\n", + " {\n", + " \"name\": \"Empty \\\\boxed{}\",\n", + " \"sample\": {\"output_text\": \"Answer: \\\\boxed{}\"},\n", + " \"item\": {\"answer\": \"42\"},\n", + " \"expected\": 0.0\n", + " }\n", + "]\n", + "all_passed = True\n", + "\n", + "for test in test_cases:\n", + " score = grade(test[\"sample\"], test[\"item\"])\n", + " passed = (score == test[\"expected\"])\n", + " all_passed = all_passed and passed\n", + " status = \"PASS\" if passed else \"FAIL\"\n", + " print(f\"{status} | {test['name']}\")\n", + " if not passed:\n", + " print(f\" Expected: {test['expected']}, Got: {score}\")\n", + "if all_passed:\n", + " print(\"ALL TESTS PASSED\")\n", + "else:\n", + " print(\"SOME TESTS FAILED\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 6. Upload Training Files\n", + "\n", + "Upload the training and validation JSONL files to Azure AI. Each file is assigned a unique ID that will be referenced when creating the fine-tuning job.\n" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Uploading training file...\n", + "Uploading validation file...\n", + "Training file ID: file-9b69d3575a1e4275bdcd7da46ea5ff19\n", + "Validation file ID: file-6e4b29f59eeb424eb70f9dfa42fec48d\n" + ] + } + ], + "source": [ + "print(\"Uploading training file...\")\n", + "with open(training_file_path, \"rb\") as f:\n", + " train_file = openai_client.files.create(file=f, purpose=\"fine-tune\")\n", + "\n", + "print(\"Uploading validation file...\")\n", + "with open(validation_file_path, \"rb\") as f:\n", + " validation_file = openai_client.files.create(file=f, purpose=\"fine-tune\")\n", + "\n", + "train_file_id = train_file.id\n", + "val_file_id = validation_file.id\n", + "\n", + "print(f\"Training file ID: {train_file_id}\")\n", + "print(f\"Validation file ID: {val_file_id}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 7. Wait for File Processing\n", + "\n", + "Azure AI needs to process the uploaded files before they can be used for fine-tuning. This step ensures the files are validated and ready for the RFT job." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Waiting for files to be processed...\n", + "Files ready!\n" + ] + } + ], + "source": [ + "print(\"Waiting for files to be processed...\")\n", + "openai_client.files.wait_for_processing(train_file_id)\n", + "openai_client.files.wait_for_processing(val_file_id)\n", + "print(\"Files ready!\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 8. Create Reinforcement Fine-Tuning Job\n", + "\n", + "Create a reinforcement fine-tuning job with your uploaded datasets and grader function. Configure hyperparameters to control the training process." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Creating Reinforcement Fine-Tuning job...\n", + "Fine-tuning job created!\n", + "Job ID: ftjob-0f1e6d94a03643bf91f167cbe940a032\n", + "Status: pending\n", + "Model: o4-mini-2025-04-16\n" + ] + } + ], + "source": [ + "print(\"Creating Reinforcement Fine-Tuning job...\")\n", + "\n", + "fine_tuning_job = openai_client.fine_tuning.jobs.create(\n", + " training_file=train_file_id,\n", + " validation_file=val_file_id,\n", + " model=model_name,\n", + " method={\n", + " \"type\": \"reinforcement\",\n", + " \"reinforcement\": {\n", + " \"grader\": grader,\n", + " \"hyperparameters\": {\n", + " \"n_epochs\": 2,\n", + " \"batch_size\": 1,\n", + " \"learning_rate_multiplier\": 1.0,\n", + " \"eval_interval\": 5,\n", + " \"eval_samples\": 2,\n", + " \"reasoning_effort\": \"medium\"\n", + " }\n", + " }\n", + " },\n", + " suffix=\"math-reasoning-rft\"\n", + ")\n", + "\n", + "print(f\"Fine-tuning job created!\")\n", + "print(f\"Job ID: {fine_tuning_job.id}\")\n", + "print(f\"Status: {fine_tuning_job.status}\")\n", + "print(f\"Model: {fine_tuning_job.model}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 9. Monitor Training Progress\n", + "\n", + "Track the status of your fine-tuning job. You can view the current status, and recent training events. Training duration varies based on dataset size, model, and hyperparameters - typically ranging from minutes to several hours." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Status: pending\n" + ] + } + ], + "source": [ + "job_status = openai_client.fine_tuning.jobs.retrieve(fine_tuning_job.id)\n", + "print(f\"Status: {job_status.status}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 10. Retrieve Fine-Tuned Model\n", + "\n", + "After the fine-tuning job succeeded, retrieve the fine-tuned model ID. This ID is required to make inference calls with your customized model." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Status: pending\n" + ] + } + ], + "source": [ + "completed_job = openai_client.fine_tuning.jobs.retrieve(fine_tuning_job.id)\n", + "\n", + "if completed_job.status == \"succeeded\":\n", + " fine_tuned_model = completed_job.fine_tuned_model\n", + " print(f\"Fine-tuned Model ID: {fine_tuned_model}\")\n", + "else:\n", + " print(f\"Status: {completed_job.status}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 11. Deploy Fine-Tuned Model\n", + "\n", + "Deploy the fine-tuned model to Azure OpenAI as a deployment endpoint. This step is required before making inference calls. The deployment uses GlobalStandard SKU with 50 capacity." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from azure.mgmt.cognitiveservices import CognitiveServicesManagementClient\n", + "from azure.mgmt.cognitiveservices.models import Deployment, DeploymentProperties, DeploymentModel, Sku\n", + "\n", + "subscription_id = os.environ.get(\"AZURE_SUBSCRIPTION_ID\")\n", + "resource_group = os.environ.get(\"AZURE_RESOURCE_GROUP\")\n", + "account_name = os.environ.get(\"AZURE_AOAI_ACCOUNT\")\n", + "\n", + "deployment_name = \"o4-mini-math-reasoning-rft\"\n", + "\n", + "with CognitiveServicesManagementClient(credential=credential, subscription_id=subscription_id) as cogsvc_client:\n", + " deployment_model = DeploymentModel(format=\"OpenAI\", name=fine_tuned_model, version=\"1\")\n", + " deployment_properties = DeploymentProperties(model=deployment_model)\n", + " deployment_sku = Sku(name=\"GlobalStandard\", capacity=200)\n", + " deployment_config = Deployment(properties=deployment_properties, sku=deployment_sku)\n", + " \n", + " print(f\"Deploying fine-tuned model: {fine_tuned_model}\")\n", + " deployment = cogsvc_client.deployments.begin_create_or_update(\n", + " resource_group_name=resource_group,\n", + " account_name=account_name,\n", + " deployment_name=deployment_name,\n", + " deployment=deployment_config,\n", + " )\n", + " \n", + " print(\"Waiting for deployment to complete...\")\n", + " deployment.result()\n", + "\n", + "print(f\"Model deployment completed: {deployment_name}\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 12. Test Fine-Tuned Model\n", + "\n", + "Test your fine-tuned model by solving a mathematical reasoning problem." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "test_problem = \"\"\"Solve for x: 3x + 7 = 22\"\"\"\n", + "\n", + "print(\"Testing fine-tuned model...\")\n", + "print(f\"Problem:{test_problem}\")\n", + "\n", + "response = openai_client.responses.create(\n", + " model=deployment_name,\n", + " input=[\n", + " {\"role\": \"system\", \"content\": \"You are a mathematical reasoning assistant. Solve problems step-by-step, showing your work clearly. Provide the final answer in \\\\boxed{} format.\"},\n", + " {\"role\": \"user\", \"content\": test_problem}\n", + " ]\n", + ")\n", + "\n", + "print(f\"Model Response:{response.output_text}\")\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "env", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.9" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Demos/RFT_Math_Reasoning/training_rft.jsonl b/Demos/RFT_Math_Reasoning/training_rft.jsonl new file mode 100644 index 0000000..a458cc6 --- /dev/null +++ b/Demos/RFT_Math_Reasoning/training_rft.jsonl @@ -0,0 +1,2661 @@ +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nCall a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers.\r\n(a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter.\r\n(b) Let K be an arbitrary integral disguisable triangle for which no smaller integral\r\ndisguisable triangle similar to it exists. Prove that at least two side lengths of K are\r\nperfect squares.", "role": "user"}], "answer": "9"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n11. A math competition problem: the probabilities of A, B, and C solving this problem independently are $\\frac{1}{a}$, $\\frac{1}{b}$, and $\\frac{1}{c}$, respectively, where $a$, $b$, and $c$ are positive integers less than 10. Now A, B, and C are solving this problem independently. If the probability that exactly one of them solves the problem is $\\frac{7}{15}$, then the probability that none of them solves the problem is", "role": "user"}], "answer": "\\dfrac{4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8 There are six cards, each with a number $1, 2, 3, 4, 5, 6$ written on it. Each time, one card is drawn, the number on it is noted, and then it is put back. This is done 4 times. The probability that the difference between the maximum and minimum numbers drawn is equal to 5 is $\\qquad$.", "role": "user"}], "answer": "\\dfrac{151"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5th Swedish 1965 Problem 2 Find all positive integers m, n such that m 3 - n 3 = 999.", "role": "user"}], "answer": "(12, 9)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n\n5. Given three positive real numbers $o, \\rho, v$, construct a triangle $A B C$ with perimeter equal to $o$, the radius of its $A$-excircle equal to $\\rho$, and the length of its $A$ altitude equal to $v$. Determine the number of solutions (non-congruent triangles) in terms of the given lengths.\n\n(Patrik Bak)\n", "role": "user"}], "answer": "0"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. In the afternoon, 5 classes need to be scheduled: Physics, Chemistry, Biology, and two self-study periods. If the first class cannot be Biology and the last class cannot be Physics, then the number of different scheduling methods is ( ) kinds.\n(A) 36\n(B) 39\n(C) 60\n(D) 78", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n9. A, B, C, and D are comparing their heights. Among them, the sum of the heights of two people is the same as the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A, and the sum of the heights of B and C is 288 cm. A's height is $\\qquad$ cm.", "role": "user"}], "answer": "139"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\\frac{1}{n}$. Similarly, the probability that a car will turn off the highway onto each of these roads is $\\frac{1}{n}$. The road by which a car leaves the highway is independent of the road by which it entered the highway.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_18848d423dc3e63c21dfg-05.jpg?height=411&width=532&top_left_y=654&top_left_x=1393)\nwhich the car leaves the highway, is independent of the road by which it entered the highway.\n\nFind the probability that a random car that has entered the highway will pass the $k$-th intersection (either drive through it or turn at it).", "role": "user"}], "answer": "\\dfrac{2k(n - k + 1) - 1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. Solve the equation $x^{\\log _{5}(0.008 x)}=\\frac{125}{x^{5}}$.", "role": "user"}], "answer": "\\dfrac{1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4.9 If $b0\n$$", "role": "user"}], "answer": "1, 2, 3, 4, 5, 6, 7"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nGiven $\\triangle A B C$ has a perimeter of 20, area of $01 \\sqrt{3}$, and $\\angle A=60^{\\circ}$. Find $\\sin A: \\sin B: \\sin C$.", "role": "user"}], "answer": "7:8:5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nB2 Points $A(-4,-2), B(2,-2)$ and $C(2,6)$ are the vertices of triangle $A B C$.\n\nA Draw triangle $A B C$ in the coordinate system.\n\nB Calculate the perimeter of triangle $A B C$.\n\nC Calculate the area of triangle $A B C$.\n\nD Circumscribe a circle around triangle $A B C$.", "role": "user"}], "answer": "(x + 1)^2 + (y - 2)^2 = 25"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n【Question 1】Calculate: $2 \\times(999999+5 \\times 379 \\times 4789)=$", "role": "user"}], "answer": "20150308"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. Simplify the expression\n\n$$\nA=\\left(1+\\frac{1+i}{2}\\right)\\left(1+\\left(\\frac{1+i}{2}\\right)^{2}\\right)\\left(1+\\left(\\frac{1+i}{2}\\right)^{4}\\right)\\left(1+\\left(\\frac{1+i}{2}\\right)^{8}\\right)\n$$\n\n( $i$ is the imaginary unit)", "role": "user"}], "answer": "\\dfrac{255(1 + i)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 3 (1) Find the domain of the function $y=\\sqrt{\\frac{1}{2}-\\log _{2}(3-x)}$;\n(2) Given that the domain of $f(x)$ is $[0,1]$, find the domain of the function $y=f\\left(\\log _{\\frac{1}{2}}(3-x)\\right)$.", "role": "user"}], "answer": "[2, \\frac{5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8. As shown in Figure $3, A C=$ $B C, A C \\perp B C$ at point $C, A B=A D=B D$, $C D=C E=D E$. If $A B=\\sqrt{2}$, then $B E=$", "role": "user"}], "answer": "1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nAt each vertex of a tetrahedron, there sits an ant. At a given moment, each of them starts to move along a randomly chosen edge and crosses over to the adjacent vertex. What is the probability that two ants meet either on the way or at the end of their journey?", "role": "user"}], "answer": "\\dfrac{25"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n7. A turtle and a rabbit are in a 1000-meter race. The rabbit took a nap 200 meters from the finish line. When it woke up, it found that the turtle was 10 meters from the finish line, so it immediately chased after and eventually both reached the finish line at the same time. If the turtle crawled at a constant speed throughout the race, and the rabbit's speed before and after the nap was the same, then, during the rabbit's nap, the turtle crawled $\\qquad$ meters.", "role": "user"}], "answer": "950"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nWe are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?", "role": "user"}], "answer": "2886"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n7. As shown in the figure, the beads on the bracelet are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming is playing a bead counting game, with the rule being: starting from bead 1, count natural numbers in a clockwise direction, but skip any number that contains the digit 7 or is a multiple of 7, and directly count the next number. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. So, when counting to 100, which bead number $\\qquad$ will it land on?", "role": "user"}], "answer": "19"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nLet $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $ \\angle B A P = 10 ^ { \\circ } , \\angle A B P = 20 ^ { \\circ } , \\angle P C A = 30 ^ { \\circ } $ and $ \\angle P A C = 40 ^ { \\circ } $. Find $ \\angle P B C $.", "role": "user"}], "answer": "60^\\circ"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n(2) In the Cartesian coordinate system $x O y$, the area of the figure enclosed by the curve $2|x|+3|y|=5$ is ( ).\n(A) $\\frac{5}{3}$\n(B) 5\n(C) $\\frac{20}{3}$\n(D) $\\frac{25}{3}$", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nGiven a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers).", "role": "user"}], "answer": "2002"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n[ Skeletons of Polyhedral Figures ]\n\nA toy factory produces wireframe cubes, with small colorful beads located at their vertices. According to the GOST standard, each cube must use beads of all eight colors (white and the seven colors of the rainbow). How many different models of cubes can the factory produce?\n\n#", "role": "user"}], "answer": "1680"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Problem Statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow 0} \\frac{6^{2 x}-7^{-2 x}}{\\sin 3 x-2 x}$", "role": "user"}], "answer": "2 \\ln 42"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nG5.3 If $R^{2000}<5^{3000}$, where $R$ is a positive integer, find the largest value of $R$.", "role": "user"}], "answer": "11"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task 18/78\n\nDetermine the greatest common divisor of all numbers $z$ that can be represented in the form $z=n^{4 m+1}-n$ with $m ; n \\in N$!", "role": "user"}], "answer": "30"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nWhich is the two-digit number that, when divided by the digit in the units place, gives a quotient of 9 and a remainder of 6?", "role": "user"}], "answer": "78"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. In $\\triangle A B C$, medians $A D$ and $C F$ intersect at point $G$. If $\\angle A F G=45^{\\circ}, \\angle A G C=60^{\\circ}$, then the degree measure of $\\angle A C F$ is ( ).\n(A) $30^{\\circ}$\n(B) $45^{\\circ}$\n(C) $60^{\\circ}$\n(D) $75^{\\circ}$", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFind the maximum positive integer $k$ such that for any positive integers $m,n$ such that $m^3+n^3>(m+n)^2$, we have\n\n$$m^3+n^3\\geq (m+n)^2+k$$\n\n[i] Proposed by Dorlir Ahmeti, Albania[/i]", "role": "user"}], "answer": "10"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n11. Choose three different digits from $0,1, \\cdots, 9$ to form a four-digit number (one of the digits can appear twice), such as 5 224. Then the total number of such four-digit numbers is $\\qquad$.", "role": "user"}], "answer": "3888"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. The three sides of a triangle $a, b, c$ are all integers, and satisfy $a b c + b c + c a + a b + a + b + c = 7$. Then the area of the triangle is equal to ().\n(A) $\\frac{\\sqrt{3}}{2}$\n(B) $\\frac{\\sqrt{2}}{4}$\n(C) $\\frac{\\sqrt{3}}{4}$\n(D) $\\frac{\\sqrt{2}}{2}$", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task Condition\n\nFind the point $M^{\\prime}$ symmetric to the point $M$ with respect to the line.\n\n$M(2 ; 1 ; 0)$\n\n$\\frac{x-2}{0}=\\frac{y+1.5}{-1}=\\frac{z+0.5}{1}$", "role": "user"}], "answer": "(2, -2, -3)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nA number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) (Cyprus) Answer: 1344", "role": "user"}], "answer": "1344"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nB5. My 24-hour digital clock displays hours and minutes only.\nFor how many different times does the display contain at least one occurrence of the digit 5 in a 24-hour period?", "role": "user"}], "answer": "450"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n17.51 In $\\triangle A B C$, $A C: C B=3: 4, \\angle C$'s exterior angle bisector intersects the extension of $B A$ at $P$ ($A$ is between $P$ and $B$), then $P A: A B$ is\n(A) $1: 3$.\n(B) $3: 4$.\n(C) $4: 3$.\n(D) $3: 1$.\n(E) $7: 1$.\n(12th American High School Mathematics Examination, 1961)", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n7. The program flowchart is shown in Figure 3. Given that the real number $x \\in [1, 9]$. Then the probability that the output $x$ is not less than 55 is ( ).\n(A) $\\frac{1}{3}$\n(B) $\\frac{2}{3}$\n(C) $\\frac{3}{8}$\n(D) $\\frac{5}{8}$", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n18. There are 30 students standing in a row, and from left to right, they are wearing hats in the order of \"red, yellow, blue, red, yellow, blue......\". The PE teacher asks the students to count off from left to right as \"1, 2, 1, 2, 1, 2.....\", and those who count off 2 step forward. Among the remaining students, those wearing red hats step back. At this point, the students are divided into 3 rows, and in the middle row, there are $\\qquad$ people wearing blue hats.", "role": "user"}], "answer": "5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. 19 If $\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{x+y+z}=1$. Then,\n(A) at least one of $x, y, z$ is equal to 1.\n(B) $x, y, z$ are all equal to 1.\n(C) $x, y, z$ are all not equal to 1.\n(D) none of the above conclusions is correct.\n(8th \"Jinyun Cup\" Junior High School Mathematics Invitational, 1991)", "role": "user"}], "answer": "A"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. Let $x_{1}$ and $x_{2}$ be the roots of the equation\n\n$$\np^{2} x^{2}+p^{3} x+1=0\n$$\n\nDetermine $p$ such that the expression $x_{1}^{4}+x_{2}^{4}$ has the smallest value.", "role": "user"}], "answer": "\\pm \\sqrt{2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n663. The grasshopper jumped from the point with coordinate 8 to the point with coordinate 17.5. Then it made another jump of the same length (in the same direction). At the point with what coordinate did it end up?", "role": "user"}], "answer": "27"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task 1 - 251211\n\nDetermine all pairs $(x ; y)$ of real numbers that satisfy the following system of equations (1), (2).\n\n$$\n\\begin{aligned}\n& x^{2}+y=1 \\\\\n& x+y^{2}=1\n\\end{aligned}\n$$", "role": "user"}], "answer": "\\left( \\frac{-1 - \\sqrt{5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe radius of the Earth measures approximately $6378 \\mathrm{~km}$ at the Equator. Suppose a wire is exactly adjusted over the Equator.\n\nThen, suppose the length of the wire is increased by $1 \\mathrm{~m}$, so that the wire and the Equator form concentric circles around the Earth. Would a man standing, an ant, or an elephant be able to pass under this wire?", "role": "user"}], "answer": "An ant"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n10) In a bag there are some marbles. Maria says: \"In the bag there are three marbles in total and they are black.\" Luca says: \"In the bag there are two black marbles and two red marbles.\" Giorgio says: \"In the bag there are only black marbles.\" Knowing that only one of the three is lying, how many marbles are in the bag?\n\n(A) one, (B) two, (C) three, (D) four, (E) it is not possible to determine the number based on the information given in the problem.", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n9. Four cousins Alan, Bob, Carl and Dan are $3,8,12$ and 14 years old, although not necessarily in that order. Alan is younger than Carl. The sum of the ages of Alan and Dan is divisible by 5. The sum of the ages of Carl and Dan is divisible by 5. What is the sum of the ages of Alan and Bob?\nA 26\nB 22\nC 17\nD 15\nE 11", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. In $\\triangle A B C$, $\\angle A=30^{\\circ}, \\angle B=105^{\\circ}$, a line $D E$ is drawn through a point $D$ on side $A C$, intersecting side $A B$ or side $B C$ at point $E$, such that $\\angle C D E=60^{\\circ}$, and $D E$ bisects the area of $\\triangle A B C$. Then $\\left(\\frac{C D}{A C}\\right)^{2}=$ $\\qquad$ -", "role": "user"}], "answer": "\\dfrac{\\sqrt{3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. (1) Use the Factor Theorem to prove that $a-b, b-c, c-a$ are all factors of $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.\n(2) Using the conclusion from (1), factorize $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.\n(3) Factorize: $(x+y+z)^{3}-x^{3}-y^{3}-z^{3}$.", "role": "user"}], "answer": "3(x + y)(y + z)(z + x)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3-ча 1. Six different colors are chosen; it is required to paint 6 faces of a cube, each in a special color from the chosen ones. In how many geometrically distinct ways can this be done? Geometrically distinct are two such colorings that cannot be made to coincide with each other by rotating the cube around its center.\n\nSolve the same problem for the case of coloring the faces of a regular dodecahedron in twelve different colors.\n\n(This problem was not solved by any of the participants in the olympiad.)", "role": "user"}], "answer": "7983360"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Problem Statement\n\nFind the point of intersection of the line and the plane.\n\n$\\frac{x+1}{3}=\\frac{y-3}{-4}=\\frac{z+1}{5}$\n\n$x+2 y-5 z+20=0$", "role": "user"}], "answer": "(2, -1, 4)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy\n\\[0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},\\]\nwhere $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$.", "role": "user"}], "answer": "447"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2 Find all integer $x$ such that $\\left|4 x^{2}-12 x-27\\right|$ is a prime number.\n\nFind all integer $x$ such that $\\left|4 x^{2}-12 x-27\\right|$ is a prime number.", "role": "user"}], "answer": "5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n10,11\n\nThe angle in the development of the lateral surface of a cone is $120^{\\circ}$. Find the angle at the vertex of the axial section of the cone.", "role": "user"}], "answer": "\\arccos \\left( \\dfrac{7"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 6 Find the number of $n$-digit numbers formed by $1,2,3$, where each of 1, 2, and 3 must appear at least once in the $n$-digit number.", "role": "user"}], "answer": "3^n - 3 \\cdot 2^n + 3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n15 . In a $3 \\times 3$ grid, fill in the numbers $1,2,3,4,5,6,7,8,9$ so that the sum of the numbers in any three squares in a row, column, or diagonal is equal.", "role": "user"}], "answer": "\\begin{array"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n\n4. How many 6-tuples $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\\right)$ are there such that each of $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ is from the set $\\{1,2,3,4\\}$ and the six expressions\n\n$$\na_{j}^{2}-a_{j} a_{j+1}+a_{j+1}^{2}\n$$\n\nfor $j=1,2,3,4,5,6$ (where $a_{7}$ is to be taken as $a_{1}$ ) are all equal to one another?\n", "role": "user"}], "answer": "40"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. In a right trapezoid $A B C D$, $A D / / B C$, $B C \\perp C D$, $E$ is the midpoint of side $A B$, $B C=C D=C E$. Then the degree measure of $\\angle B$ is ( ).\n(A) $52.5^{\\circ}$\n(B) $62.5^{\\circ}$\n(C) $60^{\\circ}$\n(D) $75^{\\circ}$", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nProblem 4. The sum of eight consecutive natural numbers is equal to 92. What are these numbers?", "role": "user"}], "answer": "15"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nIn a tournament, any two players play against each other. Each player gets one point for a win, 1/2 for a draw, and 0 points for a loss. Let $S$ be the set of the 10 lowest scores. We know that each player obtained half of their score playing against players from $S$.\n\na) What is the sum of the scores of the players in $S$?\n\nb) Determine how many participants are in the tournament.\n\nNote: Each player plays only once against each opponent.", "role": "user"}], "answer": "25"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n9. (16 points) For a given positive integer $M$, define $f_{1}(M)$ as the square of the sum of the digits of $M$. When $n>1$ and $n \\in \\mathbf{N}$, $f_{n}\\left(f_{n-1}(M)\\right)$ represents the $r_{n}$-th power of the sum of the digits of $f_{n-1}(M)$, where, when $n$ is odd, $r_{n}=2$; when $n$ is even, $r_{n}=3$. Find the value of $f_{2012}\\left(f_{2011}\\left(3^{2010}\\right)\\right)$.", "role": "user"}], "answer": "should be \\( f_{2012}(f_{2011}(3^{2010})) \\)."} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 4 Given $A(-2,2)$, and $B$ is a moving point on the ellipse $\\frac{x^{2}}{25}+\\frac{y^{2}}{16}=1$, $F$ is the left focus. When $\\mid A B \\mid +\\frac{5}{3}|B F|$ takes the minimum value, find the coordinates of point $B$.\n(1999, National High School Mathematics Competition)", "role": "user"}], "answer": "\\left( -\\dfrac{5\\sqrt{3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies $a_{1}=\\frac{7}{6}$, and for any positive integer $n$, the quadratic equation $a_{n} x^{2}-$ $a_{n+1} x+1=0$ has two roots $\\alpha_{n}, \\beta_{n}$ that satisfy $6 \\alpha_{n}-2 \\alpha_{n} \\beta_{n}+6 \\beta_{n}=3$. Then the general term formula for $\\left\\{a_{n}\\right\\}$ is $a_{n}=$ $\\qquad$", "role": "user"}], "answer": "\\dfrac{2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4.1. Mom baked four raisin buns for breakfast for her two sons. $V$ In the first three buns, she put 7, 7, 23 raisins, and some more in the fourth. It turned out that the boys ate an equal number of raisins and did not divide any bun into parts. How many raisins could Mom have put in the fourth bun? List all the options.", "role": "user"}], "answer": "37"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFind the number of ways in which three numbers can be selected from the set $\\{1,2,\\cdots ,4n\\}$, such that the sum of the three selected numbers is divisible by $4$.", "role": "user"}], "answer": "\\dfrac{n(4n -1)(2n -1)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nAlice is given a rational number $r>1$ and a line with two points $B \\neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is placed at point $X$, and the other bead is placed at $Y$, then Alice moves the chosen bead to point $X^{\\prime}$ with $\\overrightarrow{Y X^{\\prime}}=r^{k} \\overrightarrow{Y X}$. Alice's goal is to move the red bead to the point $B$. Find all rational numbers $r>1$ such that Alice can reach her goal in at most 2021 moves. Answer: All $r=(b+1) / b$ with $b=1, \\ldots, 1010$.", "role": "user"}], "answer": "r = \\frac{b+1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. Six equal circles $\\mathrm{O}, \\mathrm{O}_{1}, \\mathrm{O}_{2}, \\mathrm{O}_{3}, \\mathrm{O}_{4}$, $0_{5}$, on the surface of a sphere with radius 1, $\\odot O$ is tangent to the other five circles and $\\odot \\mathrm{O}_{1}$ is tangent to $\\odot \\mathrm{O}_{2}$, $\\odot \\mathrm{O}_{2}$ is tangent to $\\odot \\mathrm{O}_{3}$, $\\odot O_{3}$ is tangent to $\\odot O_{4}$, $\\odot O_{4}$ is tangent to $\\odot O_{5}$, $\\odot O_{5}$ is tangent to $\\odot O_{1}$, find the radius of these circles.", "role": "user"}], "answer": "\\sqrt{\\dfrac{5 - \\sqrt{5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nAs shown in Figure $3, \\odot O$ is inside the rectangle $ABCD$. Tangents are drawn from vertices $A, B, C, D$ to $\\odot O$, with points of tangency being $A_{1}, B_{1}, C_{1}, D_{1}$, respectively. If $AA_{1}=3, BB_{1}=4$, $CC_{1}=5$, find the length of $DD_{1}$.", "role": "user"}], "answer": "3\\sqrt{2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe Torricelli experiment is to be performed with ether instead of mercury. The room temperature in one case is $t_{1}=$ $30^{\\circ}$, and in the other case $t_{2}=-10^{\\circ}$, the air pressure in both cases is equal to $750 \\mathrm{~mm}$ of mercury pressure. The density of ether is $\\sigma_{1}=0.715$, that of mercury is $\\sigma_{2}=13.6$; the vapor pressure of ether is $30^{\\circ}$, $634.8 \\mathrm{~mm}$ of mercury column pressure at $10^{\\circ}$, and $114.72 \\mathrm{~mm}$ of mercury column pressure at $-10^{\\circ}$. How high is the ether column at the two temperatures?", "role": "user"}], "answer": "12084\\ \\text{mm"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nCondition of the problem\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow 1}\\left(\\frac{x^{3}-1}{x-1}\\right)^{\\frac{1}{x^{2}}}$", "role": "user"}], "answer": "3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n$$\n\\begin{array}{l}\n\\text { Three. (20 points) (1) The quadratic function } \\\\\nf(x)=a x^{2}+b x+c(a, b, c \\in \\mathbf{R}, a \\neq 0)\n\\end{array}\n$$\n\nsatisfies\n(i) For $x \\in \\mathbf{R}$, there is $4 x \\leqslant f(x) \\leqslant \\frac{1}{2}(x+2)^{2}$\n\nalways holds;\n(ii) $f(-4+2 \\sqrt{3})=0$.\nFind $f(x)$.\n(2) Let $f_{1}(x)=\\frac{3}{2+x}, f_{n+1}(x)=f_{1}\\left(f_{n}(x)\\right)$, find $f_{2009}(0)$.", "role": "user"}], "answer": "\\dfrac{3^{2010"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. Let $S=\\left\\{(x, y) \\mid x^{2}-y^{2}=\\right.$ odd, $\\left.x, y \\in R\\right\\}, T=\\{(x, y) \\mid$ $\\left.\\sin \\left(2 \\pi x^{2}\\right)-\\sin \\left(2 \\pi y^{2}\\right)=\\cos \\left(2 \\pi x^{2}\\right)-\\cos \\left(2 \\pi y^{2}\\right), x, y \\in R\\right\\}$. Then\n(A) $S \\subset T$;\n(B) $T \\subset S$;\n(C) $S=T$;\n(D) $S \\cap T=\\Phi$.", "role": "user"}], "answer": "A"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. In trapezoid $A B C D$, $A B / / C D$, the base angles $\\angle D A B=36^{\\circ}, \\angle C B A=54^{\\circ}, M$ and $N$ are the midpoints of sides $A B$ and $C D$, respectively. If the lower base $A B$ is exactly 2008 units longer than the upper base $C D$, then the line segment $M N=$ $\\qquad$", "role": "user"}], "answer": "1004"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe seventh grades of our school competed in collecting caps from PET bottles. Class A collected half of what classes B and C collected together, class B collected a third of what classes A and C collected together, and class C collected 150 caps.\n\nDetermine how many caps these three classes collected in total.\n\n(M. Volfová)", "role": "user"}], "answer": "360"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nIn a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$\n$\\mathrm{(A)}\\ 12\\qquad \\mathrm{(B)}\\ 17\\qquad \\mathrm{(C)}\\ 20\\qquad \\mathrm{(D)}\\ 25\\qquad \\mathrm{(E)}\\ 35$", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n15. Given $f(x)=x^{2}+c$, and $f(f(x))=f\\left(x^{2}+1\\right)$.\n(1) Let $g(x)=f(f(x))$, find the analytical expression of the function $g(x)$;\n(2) Let $\\varphi(x)=g(x)-\\lambda f(x)$, try to find the value of the real number $\\lambda$ such that $\\varphi(x)$ is a decreasing function on $(-\\infty,-1]$ and an increasing function on $[-1,0)$.", "role": "user"}], "answer": "4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?\n$\\text{(A)}\\ 162 \\qquad \\text{(B)}\\ 172 \\qquad \\text{(C)}\\ 173 \\qquad \\text{(D)}\\ 174 \\qquad \\text{(E)}\\ 222$", "role": "user"}], "answer": "A"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. Given are the sides $a = BC$ and $b = CA$ of triangle $ABC$. Determine the length of the third side if it is equal to the length of the corresponding height. For which values of $a$ and $b$ does the problem have a solution?", "role": "user"}], "answer": "\\sqrt{\\dfrac{a^2 + b^2 + 2\\sqrt{3a^2b^2 - a^4 - b^4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task 15/85\n\nLet the numbers $a ; b ; c$ represent the side lengths of a triangle with perimeter $U$, and let $a^{2}$; $b^{2} ; c^{2}$ represent the side lengths of another triangle with perimeter $U^{\\prime}$. Determine the lower bound of the ratio $U^{2}: U^{\\prime}$!", "role": "user"}], "answer": "2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n7. Let the edge length of a regular tetrahedron be $2 \\sqrt{6}$, and let a sphere be constructed with the center $O$ of the tetrahedron as its center. The total length of the curves formed by the intersection of the sphere's surface with the four faces of the tetrahedron is $4 \\pi$. Then the radius of the sphere $O$ is $\\qquad$", "role": "user"}], "answer": "\\dfrac{\\sqrt{5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe graph shows the fuel used per $100 \\mathrm{~km}$ of driving for five different vehicles. Which vehicle would travel the farthest using 50 litres of fuel?\n(A) $U$\n(B) $V$\n(C) $W$\n(D) $X$\n(E) $Y$\n\n![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-059.jpg?height=431&width=380&top_left_y=1297&top_left_x=1271)", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 2.65. Calculate the volume of the body formed by the rotation around the $O Y$ axis of the curvilinear trapezoid bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$ and $y_{3}=0$.", "role": "user"}], "answer": "3\\pi"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFor two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\\ell(x)$ such that $P(x)=Q(\\ell(x))$ for all real $x$. How many such linear functions $\\ell(x)$ can exist?\n\n[i](A. Golovanov)[/i]", "role": "user"}], "answer": "2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. A car is driving at a constant speed in one direction along a straight road, near which there are two houses. At noon, when the car had not yet reached the houses, the sum of the distances from it to these houses was 10 km. After 10 minutes, when the car had already passed both houses, it turned out that the sum of the distances from it to the houses was again 10 km. What is the speed of the car? (Problem author - I. Rubanov)", "role": "user"}], "answer": "60"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n15. Xiaohua wrote a computer program. One minute after the program runs, some bubbles first appear on the computer screen, and at each subsequent integer minute, the same number of new bubbles appear. Half a minute after the 11th appearance of bubbles, one bubble bursts. Thereafter, every minute, more bubbles burst, with the number increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles burst...). At a certain moment, the total number of burst bubbles is exactly equal to the total number of bubbles that have appeared on the screen, meaning all bubbles have disappeared at this moment. How many bubbles were there at most on the screen at any given time during the entire process of the program running?", "role": "user"}], "answer": "1026"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nA triangular prism has a volume of $120 \\mathrm{~cm}^{3}$. Two edges of the triangular faces measure $3 \\mathrm{~cm}$ and $4 \\mathrm{~cm}$, as shown. The height of the prism, in $\\mathrm{cm}$, is\n(A) 12\n(B) 20\n(C) 10\n(D) 16\n(E) 8\n\n![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-050.jpg?height=358&width=336&top_left_y=645&top_left_x=1334)", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n$18 \\cdot 110$ As shown, $A B C D$ is an isosceles trapezoid, $A D$ $=B C=5, A B=4, D C=10$, point $C$ is on $D F$, and $B$ is the midpoint of the hypotenuse of the right triangle $\\triangle D E F$. Then $C F$ equals\n(A) 3.25 .\n(B) 3.5 .\n(C) 3.75 .\n(D) 4.0 .\n(E) 4.25 .", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. Given 2117 cards, on which natural numbers from 1 to 2117 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done?", "role": "user"}], "answer": "22386"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. (6 points) A certain duplicator can print 3600 sheets of paper per hour, so printing 240 sheets of paper requires $\\qquad$ minutes.", "role": "user"}], "answer": "4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. At the food station, four track and field athletes drank all the cola-loc lemonade. If athlete Bystrov had drunk only half as much, a tenth of the lemonade would have remained. If, in addition, athlete Shustrov had also drunk half as much, an eighth of the lemonade would have remained. If, in addition to them both, athlete Vostrov had also drunk half as much, a third of the lemonade would have remained. What fraction of the lemonade would remain if only athlete Pereskochizaborov had drunk half as much?", "role": "user"}], "answer": "\\dfrac{1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1 $\\left[\\begin{array}{ll}\\text { Common fractions }\\end{array}\\right]$\n\nWhich of the three fractions is the largest: $3 / 4, 4 / 5$ or $5 / 6$?", "role": "user"}], "answer": "\\dfrac{5"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nIn the given operation, the letters $a, b$, and $c$ represent distinct digits and are different from 1. Determine the values of $a, b$, and $c$.\n\n$$\n\\begin{array}{r}\na b b \\\\\n\\times \\quad c \\\\\n\\hline b c b 1\n\\end{array}\n$$", "role": "user"}], "answer": "c=7"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nCondition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{\\cos \\ln 7 \\cdot \\sin ^{2} 7 x}{7 \\cos 14 x}\n$$", "role": "user"}], "answer": "\\dfrac{\\cos(\\ln 7) \\sin(14x)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n10.1. (12 points) Find the minimum value of the function $f(x)=x^{2}+3 x+\\frac{6}{x}+\\frac{4}{x^{2}}-1$ on the ray $x>0$.", "role": "user"}], "answer": "3 + 6\\sqrt{2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 6 Xiao Ming divides a cube with an edge length of 4 into 29 smaller cubes with integer edge lengths. Then the number of smaller cubes with an edge length of 1 is ( ).\n(A) 22\n(B) 23\n(C) 24\n(D) 25", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThe number of solutions $(P, Q)$ of the equation $\\frac{P}{Q}-\\frac{Q}{P}=\\frac{P+Q}{P Q}$, where $P$ and $Q$ are integers from 1 to 9 inclusive, is\n(A) 1\n(B) 8\n(C) 16\n(D) 72\n(E) 81", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n[ Auxiliary area. The area helps to solve the problem_] Heron's formula\n\nThe sides of the triangle are 13, 14, and 15. Find the radius of the circle that has its center on the middle side and touches the other two sides.", "role": "user"}], "answer": "6"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFind all subsets $A$ of $\\left\\{ 1, 2, 3, 4, \\ldots \\right\\}$, with $|A| \\geq 2$, such that for all $x,y \\in A, \\, x \\neq y$, we have that $\\frac{x+y}{\\gcd (x,y)}\\in A$.\n\n[i]Dan Schwarz[/i]", "role": "user"}], "answer": "\\{a, a(a-1)\\"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. The numbers $a, b, c, d$ are in arithmetic progression, while the numbers $a+1, b-1, c-1, d+3$ are in geometric progression. Find the numbers.\n\nGrade 9 Manual\n\nAll problems are mandatory.\n\nEach problem is scored from 0 to 7 points.\n\nWorking time 3 hours.\n\n## Grading Rubric OML Grade 9, 2014", "role": "user"}], "answer": "13"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n285. The random variable $X$ is given by the probability density function $f(x)=2 \\cos 2x$ in the interval $(0, \\pi / 4)$; outside this interval, $f(x)=0$. Find: a) the mode; b) the median of $X$.", "role": "user"}], "answer": "\\dfrac{\\pi"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task 5 - 200835\n\nTwo rays $s_{1}$ and $s_{2}$, which originate from a point $S$ and form a right angle with each other, are intersected by two parallel lines $g$ and $h$. The line $g$ intersects $s_{1}$ at $A$ and $s_{2}$ at $C$, the line $h$ intersects $s_{1}$ at $B$ and $s_{2}$ at $D$. Furthermore, $\\overline{S B}=5 \\mathrm{~cm}$, and the area of triangle $S A C$ is exactly $36 \\%$ of the area of triangle $S B D$.\n\nDetermine the length of the segment $S A$ from these conditions!", "role": "user"}], "answer": "3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 16 Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers, find the smallest possible values of these two squares $\\left(\\mathrm{IMO}_{37-4}\\right)$.", "role": "user"}], "answer": "231361"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n22. Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1 . When the smallest number is removed, the average is increased by 1 . When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20 . Find the product of the largest and the smallest numbers.", "role": "user"}], "answer": "375"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. For which integer values of $a$ and $b$ does the system\n\n$$\n\\left\\{\\begin{array}{l}\nm^{2}+n^{2}=b \\\\\n\\frac{m^{n}-1}{m^{n}+1}=a\n\\end{array}\\right.\n$$\n\nhave a solution (for $m$ and $n$) in the set $\\mathbb{Z}$ of integers.", "role": "user"}], "answer": "(a, b) = (3, 5)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 10 (1994 National Training Team Selection Test) Find all quadruples of positive integers $a, b, c, d$ such that the product of any three numbers divided by the remaining one leaves a remainder of 1.", "role": "user"}], "answer": "(2, 3, 11, 13)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n# Task 11.1. (10 points)\n\nGiven a sequence of numbers $x_{1}, x_{2}, \\ldots$, such that $x_{1}=79$ and $x_{n}=\\frac{n}{x_{n-1}}$ for all $n>1$. How many zeros does the number equal to the product $x_{1} x_{2} \\ldots x_{2018}$ end with?", "role": "user"}], "answer": "250"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. In $\\triangle A B C$, $A>B>C$, if $2 \\cos 2 B-$ $8 \\cos B+5=0, \\tan A+\\tan C=3+\\sqrt{3}$, and the altitude $C D$ from $C$ to $A B$ is $2 \\sqrt{3}$, then the area of $\\triangle A B C$ is", "role": "user"}], "answer": "12 - 4\\sqrt{3"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFor each $k$, $\\mathcal{C}_k$ is biased so that, when tossed, it has probability $\\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.", "role": "user"}], "answer": "\\dfrac{n"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. Solve the equation\n\n$$\n\\sqrt{x^{2}+x}+\\sqrt{1+\\frac{1}{x^{2}}}=\\sqrt{x+3}\n$$", "role": "user"}], "answer": "-1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nZhendarov R.G.\n\nIn the cells of a $4 \\times 4$ table, numbers are written such that the sum of the neighbors of each number is 1 (cells are considered neighbors if they share a side).\n\nFind the sum of all the numbers in the table.\n\n#", "role": "user"}], "answer": "6"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8.072. $(\\cos 6 x-1) \\operatorname{ctg} 3 x=\\sin 3 x$.\n\n8.072. $(\\cos 6 x-1) \\cot 3 x=\\sin 3 x$.", "role": "user"}], "answer": "x = \\dfrac{4\\pi"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n10. Arrange all positive integers that are coprime with 70 in ascending order, the 2017th term of this sequence is $\\qquad$ .", "role": "user"}], "answer": "5881"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nThree. (50 points)\nThere are 21 points on a circle. Among all the arcs with these 21 points as endpoints, what is the minimum number of arcs that do not exceed $120^{\\circ}$?", "role": "user"}], "answer": "11"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. One light-year is nearly $6 \\times 10^{12}$ miles. In 2016 , the Hubble Space Telescope set a new cosmic record, observing a galaxy 13.4 thousand million light-years away.\nRoughly how many miles is that?\nA $8 \\times 10^{20}$\nB $8 \\times 10^{21}$\nC $8 \\times 10^{22}$\nD $8 \\times 10^{23}$\nE $8 \\times 10^{24}$", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task 1 - 110721\n\nDetermine all three-digit natural numbers that are simultaneously divisible by 2, 3, 4, 6, 7, 8, 9, 12, and 14!", "role": "user"}], "answer": "504"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n(4) Let $x, y, z \\in \\mathbf{R}^{+}, x+y+z=1$, then the maximum value of the function $f(x, y, z)=x y^{2} z^{3}$ is $\\qquad$", "role": "user"}], "answer": "\\dfrac{1"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 12 Let $n$ be a fixed integer, $n \\geqslant 2$.\n(1) Determine the smallest constant $c$ such that the inequality\n$$\n\\sum_{1 \\leqslant ic$, then $\\overline{\\mathrm{abc}}=$ $\\qquad$ .", "role": "user"}], "answer": "452"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n11. On Qingming Festival, the students take a bus to the martyrs' cemetery to pay respects. If the bus increases its speed by one-fifth after driving for 1 hour, it can arrive 20 minutes earlier than the scheduled time; if the bus first travels 72 kilometers at the original speed and then increases its speed by one-third, it can arrive 30 minutes earlier than the scheduled time. How many kilometers is it from the school to the martyrs' cemetery?", "role": "user"}], "answer": "216"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8. In triangle $A B C$ with $\\angle B=120^{\\circ}$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn. Segment $A_{1} B_{1}$ intersects the angle bisector $C C_{1}$ at point M. Find $\\angle C B M$.\n\n#", "role": "user"}], "answer": "30^\\circ"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nLet $p$ and $q$ be different prime numbers. How many divisors does the number have:\na) $p q ;$\n\nb) $p^{2} q ;$\n\nc) $p^{2} q^{2}$;\n\nd) $p^{m} q^{n}$ ?", "role": "user"}], "answer": "23"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. From the vertex of the obtuse angle $A$ of triangle $A B C$, the altitude $A D$ is dropped. A circle with center $D$ and radius $D A$ is drawn, intersecting sides $A B$ and $A C$ again at points $M$ and $N$ respectively. Find $A C$, if $A B=c, A M=m$ and $A N=n$.", "role": "user"}], "answer": "\\dfrac{cm"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8.4. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galia had moved two seats to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. Which seat did Anya occupy before she got up?", "role": "user"}], "answer": "4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. Find the largest natural number $n$ with the following property: for any odd prime $p$ less than $n$, the difference $n-p$ is also a prime number.\n\n(I. Akulich)", "role": "user"}], "answer": "10"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nLet $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \\times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy. ", "role": "user"}], "answer": "(m, n) \\text{ where"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. (3 points) In a puddle, there live three types of amoebas: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning, there were 47 red, 40 blue, and 53 yellow amoebas in the puddle, and by evening, only one amoeba remained. What color is it?\n\n#", "role": "user"}], "answer": "blue"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. Determine all positive real numbers $a, b, c, d$ for which\n1) $16abcd = \\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)(a b c+b c d+c d a+d a b)$ and\n2) $2 a b+2 c d+a c+a d+b c+b d=8$.", "role": "user"}], "answer": "(1, 1, 1, 1)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n3. Given the regions $x^{2}+y^{2} \\leqslant 1$ and $[x]^{2}+[y]^{2}=1$ with areas $S_{1}$ and $S_{2}$ respectively (where $[x]$ denotes the greatest integer less than or equal to $x$). Then ( ).\n(A) $S_{1}>S_{2}$\n(B) $S_{1}=S_{2}$\n(C) $S_{1}b>c$ and that the units digit of the printed product is the correct digit 6. Restore the true value of the product.", "role": "user"}], "answer": "2235542836"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n8.1. In a circle, 58 balls of two colors - red and blue - are arranged. It is known that the number of triples of consecutive balls, among which there are more red ones, is the same as the number of triples with a majority of blue ones. What is the smallest number of red balls that could be present?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_6c94238ddc8917ebef5dg-8.jpg?height=891&width=897&top_left_y=231&top_left_x=591)", "role": "user"}], "answer": "29"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n9. (16 points) Given that the odd function $f(x)$ is increasing on the interval $(-\\infty, 0)$, and $f(-2)=-1, f(1)=0$, when $x_{1}>0, x_{2}>0$, we have $f\\left(x_{1} x_{2}\\right)=f\\left(x_{1}\\right)+f\\left(x_{2}\\right)$, find the solution set of the inequality $\\log _{2}|f(x)+1|<0$.", "role": "user"}], "answer": "(-4, -2) \\cup (-2, -1) \\cup (1, 2) \\cup (2, 4)"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. Given that $A_{1}, B_{1}, C_{1}, D_{1}$ are points on the four sides of square $A B C D$. The following are two propositions about it:\n(1) If $A_{1} B_{1} C_{1} D_{1}$ is a rectangle with unequal adjacent sides, then the rectangle $A_{1} B_{1} C_{1} D_{1}$ must have a side parallel to one of the diagonals of the square $A B C D$;\n(2) If $A_{1} B_{1} C_{1} D_{1}$ is an isosceles trapezoid, then the isosceles trapezoid $A_{1} B_{1} C_{1} D_{1}$ must have a base parallel to one of the diagonals of the square $A B C D$.\nThen, ( ).\n(A) (1) and (2) are both true propositions\n(B) (1) and (2) are both false propositions\n(C) (1) is a false proposition, (2) is a true proposition\n(D) (1) is a true proposition, (2) is a false proposition", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. (10 points) Rearrange the 37 different natural numbers $1,2,3, \\cdots, 37$ into a sequence, denoted as $a_{1}, a_{2}, \\cdots, a_{37}$, where $a_{1}=37, a_{2}=1$, and ensure that $a_{1}+a_{2}+\\cdots+a_{k}$ is divisible by $a_{k+1}$ $(k=1,2, \\cdots, 36)$. Find $a_{3}=? a_{37}=?$", "role": "user"}], "answer": "19"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n6. (10 points) An exam has 3 questions. After the exam, four friends check their answers and find that they got $3, 2, 1,$ and $0$ questions right, respectively. At this moment, the teacher asks: How did you do on the exam? Each of them says 3 sentences (as follows):\n甲: I got two questions right, and I did better than 乙, 丙 did worse than 丁.\n乙: I got all questions right, 丙 got none right, 甲 did worse than 丁.\n丙: I got one question right, 丁 got two questions right, 乙 did worse than 甲.\n丁: I got all questions right, 丙 did worse than me, 甲 did worse than 乙.\nIf each person tells as many true statements as the number of questions they got right, and let 甲, 乙, 丙, 丁 get $A, B, C, D$ questions right, respectively, then the four-digit number $\\overline{\\mathrm{ABCD}}=$", "role": "user"}], "answer": "1203"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n13. If $x y=5$ and $x^{2}+y^{2}=21$, compute $x^{4}+y^{4}$.", "role": "user"}], "answer": "391"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 2. Compute the integral\n\n$$\n\\int_{C}\\left(z^{2}+z \\bar{z}\\right) d z\n$$\n\nwhere $C-$ is the arc of the circle $\\{z \\mid=1(0 \\leqslant \\arg z \\leqslant \\pi)$.", "role": "user"}], "answer": "-\\dfrac{8"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. The number of real solutions to the equation $\\left|x^{2}-3 x+2\\right|+\\left|x^{2}+2 x-3\\right|=11$ is ( ).\n(A) 0\n(B) 1\n(C) 2\n(D) 4", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n4. The bases $AB$ and $CD$ of the trapezoid $ABCD$ are extended on both sides. The bisectors of the exterior angles of the trapezoid at vertices $A$ and $D$ intersect at point $M$, and the bisectors of the exterior angles at vertices $B$ and $C$ intersect at point $N$. Determine the perimeter of trapezoid $ABCD$ if $MN=2k$.", "role": "user"}], "answer": "4k"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n5. Given $x, y \\in [0,+\\infty)$ and satisfy $x^{3}+y^{3}+3 x y=1$.\nThen the maximum value of $x^{2} y$ is $\\qquad$", "role": "user"}], "answer": "\\dfrac{4"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nPoints $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ and $\\overline{AG}$ are parallel. Find $HC/JE$.\n$\\text{(A)}\\ 5/4 \\qquad \\text{(B)}\\ 4/3 \\qquad \\text{(C)}\\ 3/2 \\qquad \\text{(D)}\\ 5/3 \\qquad \\text{(E)}\\ 2$", "role": "user"}], "answer": "D"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n9. For a cube, either one of its edges is painted red, or all are painted black, with each face having at least one edge painted black. The minimum number of black edges is ( ).\n(A) 2.\n(B) 3.\n(C) 4.\n(D) 5.\n(E) 6.", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nExample 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.\n\n untranslated text:\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 \n\n translated text:\nExample 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.\n\nThe text above has been translated into English, preserving the original text's line breaks and format.", "role": "user"}], "answer": "8"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nLet $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \\frac{p^2q^2}{6}\\left(a - \\frac{b}{c} \\right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.", "role": "user"}], "answer": "7561"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n34th Swedish 1994 Problem 6 Let N be the set of non-negative integers. The function f:N→N satisfies f(a+b) = f(f(a)+b) for all a, b and f(a+b) = f(a)+f(b) for a+b < 10. Also f(10) = 1. How many three digit numbers n satisfy f(n) = f(N), where N is the \"tower\" 2, 3, 4, 5, in other words, it is 2 a , where a = 3 b , where b = 4 5 ?", "role": "user"}], "answer": "100"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n## Task $7 / 84$\n\nA man says: \"The birth year of my grandson is the product $x y$ of two natural numbers $x$ and $y$. In the year $x^{2}$, he will be $x$ years old.\"\n\nIn which year was the grandson born?", "role": "user"}], "answer": "1980"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n87 Given the complex sequence $\\left\\{a_{n}\\right\\}$ with the general term:\n$$\na_{n}=(1+i)\\left(1+\\frac{i}{\\sqrt{2}}\\right)\\left(1+\\frac{i}{\\sqrt{3}}\\right) \\cdots\\left(1+\\frac{i}{\\sqrt{n}}\\right)\n$$\n\nthen $\\left|a_{n}-a_{n+1}\\right|=$\nA. $\\frac{\\sqrt{2}}{2}$\nB. $\\sqrt{2}$\nC. 1\nD. 2", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2.2.5 ** If $x \\in\\left[-\\frac{5 \\pi}{12},-\\frac{\\pi}{3}\\right]$, then the maximum value of $y-\\tan \\left(x+\\frac{2 \\pi}{3}\\right)-\\tan \\left(x+\\frac{\\pi}{6}\\right)+$ $\\cos \\left(x+\\frac{\\pi}{6}\\right)$ is ( ) .\n(A) $\\frac{12}{5} \\sqrt{2}$\n(B) $\\frac{11}{6} \\sqrt{2}$\n(C) $\\frac{11}{6} \\sqrt{3}$\n(D) $\\frac{12}{5} \\sqrt{3}$", "role": "user"}], "answer": "C"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nFind all positive integers $k$ for which the equation: $$ \\text{lcm}(m,n)-\\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\\neq n$.\n", "role": "user"}], "answer": "2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nTask 4. (20 points) For the numerical sequence $\\left\\{x_{n}\\right\\}$, all terms of which, starting from $n \\geq 2$, are distinct, the relation $x_{n}=\\frac{x_{n-1}+398 x_{n}+x_{n+1}}{400}$ holds. Find $\\sqrt{\\frac{x_{2023}-x_{2}}{2021} \\cdot \\frac{2022}{x_{2023}-x_{1}}}+2021$.", "role": "user"}], "answer": "\\sqrt{2022"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n2. In $\\triangle A B C$, if $A C=\\sqrt{2}, A B=2$, and $\\frac{\\sqrt{3} \\sin A+\\cos A}{\\sqrt{3} \\cos A-\\sin A}=\\tan \\frac{5 \\pi}{12}$, then $B C=$ $\\qquad$", "role": "user"}], "answer": "\\sqrt{2"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n1. A pit in the shape of a rectangular prism has been dug. Its depth is 12 times greater than its length. The length of the pit is $\\frac{3}{2}$ times its width. The sum of the numerical value of the volume of the pit and the area of its base is $\\frac{7}{6}$. Calculate the depth of the pit.", "role": "user"}], "answer": "6"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\nIf a dealer could get his goods for $8$% less while keeping his selling price fixed, \nhis profit, based on cost, would be increased to $(x+10)$% from his present profit of $x$%, which is\n$\\textbf{(A) }12\\%\\qquad \\textbf{(B) }15\\%\\qquad \\textbf{(C) }30\\%\\qquad \\textbf{(D) }50\\%\\qquad \\textbf{(E) }75\\%$", "role": "user"}], "answer": "B"} +{"messages": [{"content": "You are a mathematical reasoning expert. Solve problems with detailed step-by-step thinking and provide final answers in \\boxed{} format. Show all intermediate calculations and explain your reasoning clearly.\n\n26. Given that $y=f(x)$ is a decreasing function on the domain $(-1,1)$, and $f(1-a)