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"capability_description": "The math_mathematics_modeling_real_world capability consists of 1500 real-world mathematical modeling problems. Each problem requires the application of mathematical concepts to formulate and solve real-life scenarios, emphasizing critical thinking and problem-solving skills.",
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"capability_domain": "math",
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"capability_instructions": "f\"\"\"Solve the following real-world modeling problem step by step. The last line of your response should be of the form \"ANSWER: $ANSWER\" (without quotes) where $ANSWER is the answer to the problem.\n\nProblem: {t[\"problem\"]}\n\nRemember to put your answer on its own line at the end in the form \"ANSWER:$ANSWER\" (without quotes) where $ANSWER is the answer to the problem, and you do not need to use a \\boxed command.\"\"\"",
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"capability_data": [
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{
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"id": "1",
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"problem": "A company wants to determine the optimal number of units to produce in order to maximize profit. The profit function is given by P(x) = -2x^2 + 40x - 150, where x is the number of units produced. How many units should the company produce to maximize profit?",
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"answer": "10"
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},
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{
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"id": "2",
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"problem": "A farmer has 240 meters of fencing to create a rectangular pen. What dimensions will maximize the area of the pen?",
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"answer": "60 by 60"
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},
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{
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"id": "3",
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"problem": "A car rental company charges a flat fee of $50 plus $20 per day. If a customer has a budget of $200, how many days can they rent a car?",
"problem": "A company wants to determine the optimal number of units to produce in order to maximize profit. The profit function is given by P(x) = -2x^2 + 40x - 150, where x is the number of units produced. How many units should the company produce to maximize profit?",
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"answer": "10"
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},
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"2": {
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"problem": "A farmer has 240 meters of fencing to create a rectangular pen. What dimensions will maximize the area of the pen?",
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"answer": "60 by 60"
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},
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"3": {
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"problem": "A car rental company charges a flat fee of $50 plus $20 per day. If a customer has a budget of $200, how many days can they rent a car?",
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"answer": "7"
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}
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}
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@staticmethod
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defget_instructions(t: dict) ->str:
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returnf"""Solve the following real-world modeling problem step by step. The last line of your response should be of the form "ANSWER: $ANSWER" (without quotes) where $ANSWER is the answer to the problem.\n\nProblem: {t["problem"]}\n\nRemember to put your answer on its own line at the end in the form "ANSWER:$ANSWER" (without quotes) where $ANSWER is the answer to the problem, and you do not need to use a \\boxed command."""
"class": '```python\nclass Capability:\n @staticmethod\n def repr_tasks() -> dict[str, dict]:\n return {\n "1": {\n "problem": "Prove that the number of ways to choose 2 elements from a set of n elements is equal to the number of ways to choose n-2 elements from the same set.",\n "answer": "\\\\binom{n}{2} = \\\\binom{n}{n-2}"\n },\n "2": {\n "problem": "Show that for any positive integer n, the sum of the first n odd numbers equals n^2.",\n "answer": "1 + 3 + 5 + ... + (2n-1) = n^2"\n },\n "3": {\n "problem": "Demonstrate that \\sum_{k=0}^{n} \\binom{n}{k} = 2^n using a combinatorial argument.",\n "answer": "\\\\sum_{k=0}^{n} \\binom{n}{k} = 2^n"\n }\n}\n\n @staticmethod\n def get_instructions(t: dict) -> str:\n return f"""Provide a combinatorial proof for the following problem. The last line of your response should be of the form "ANSWER: $ANSWER" (without quotes) where $ANSWER is your proof or explanation.\\n\\nProblem: {t["problem"]}\\n\\nRemember to put your proof or explanation on its own line at the end in the form "ANSWER:$ANSWER" (without quotes) where $ANSWER is your proof or explanation."""\n\n @staticmethod\n def score(t: dict, submission: str) -> float | None:\n return 1.0 if submission.lower().strip() == t["answer"].lower().strip() else 0.0\n```',
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