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--- |
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base_model: deepseek-ai/deepseek-math-7b-base |
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tags: |
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- alignment-handbook |
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- generated_from_trainer |
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- aimo |
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widget: |
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- example_title: Math problem |
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messages: |
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- role: user |
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content: >- |
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For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ |
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have two distinct integer roots? |
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output: |
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text: >- |
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### Solution: 1. For the polynomial \\( x^2 + kx + 36 \\) to have two |
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distinct integer roots, let's denote these roots by \\( r_1 \\) and \\( |
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r_2 \\).\n\n2. According to Vieta's formulas, the sum of the roots \\( r_1 |
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+ r_2 \\) is equal to \\(-k\\), and the product of the roots \\( r_1 |
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\\cdot r_2 \\) is equal to 36.\n\n3. To find the distinct integer pairs |
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\\((r_1, r_2)\\) whose product is 36, we need to determine all pairs of |
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integers \\((r_1, r_2)\\) such that \\( r_1 \\cdot r_2 = 36 \\) and \\( |
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r_1 \\neq r_2 \\).\n\n4. From the equation \\( r_1 + r_2 = -k \\), for |
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each valid pair \\((r_1, r_2)\\), we can compute the corresponding value |
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of \\( k \\).\n\n5. Finally, since we need the polynomial to have two |
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distinct integer roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\) |
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are distinct.\n\nLet's start by finding all pairs \\((r_1, r_2)\\) such |
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that \\( r_1 \\cdot r_2 = 36 \\). We'll then determine the values of \\( k |
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\\) and ensure the roots are distinct.\n```python\nimport itertools\n\n# |
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Find all pairs (r1, r2) such that r1 * r2 = 36\nproduct_36 = |
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36\nfactor_pairs = []\n\nfor i in range(1, product_36 + 1):\n if |
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product_36 % i == 0:\n pair = (i, product_36 // i)\n if pair[0] != |
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pair[1]: # Ensure distinct pairs\n factor_pairs.append(pair)\n\n# |
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Calculate k for each pair and ensure distinct integer |
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roots\nvalid_k_values = set()\nfor r1, r2 in factor_pairs:\n if r1 != |
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r2:\n k = -(r1 + r2)\n |
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valid_k_values.add(k)\n\nprint((len(valid_k_values), |
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sorted(valid_k_values)))\n```\n```output\n(4, [-37, -20, -15, |
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-13])\n```\nThe distinct integer values of \\( k \\) that make the |
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polynomial \\( x^2 + kx + 36 \\) have two distinct integer roots are |
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\\(-37, -20, -15, \\text{and} -13\\).\n\nTherefore, the number of such |
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values of \\( k \\) is:\n\n\\[\n\\boxed{4}\n\\]\n |
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pipeline_tag: text-generation |
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model-index: |
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- name: NuminaMath-7B-TIR-GPTQ |
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results: [] |
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license: apache-2.0 |
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language: |
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- en |
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--- |
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<!-- This model card has been generated automatically according to the information the Trainer had access to. You |
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should probably proofread and complete it, then remove this comment. --> |
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<img src="https://huggingface.co/AI-MO/NuminaMath-7B-TIR/resolve/main/thumbnail.png" alt="Numina Logo" width="800" style="margin-left:'auto' margin-right:'auto' display:'block'"/> |
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# Model Card for NuminaMath 7B TIR GPTQ |
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NuminaMath is a series of language models that are trained to solve math problems using tool-integrated reasoning (TIR). NuminaMath 7B TIR won the first progress prize of the [AI Math Olympiad (AIMO)](https://aimoprize.com), with a score of 29/50 on the public and private tests sets. |
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This model is an 8-bit version of [`AI-MO/NuminaMath-7B-TIR`](https://huggingface.co/AI-MO/NuminaMath-7B-TIR), which we quantized with [AutoGPTQ](https://github.com/AutoGPTQ/AutoGPTQ) to run fast inference in the Kaggle submissions. Please consult the original model card for more details. |
