64 lines
3.4 KiB
Markdown
64 lines
3.4 KiB
Markdown
---
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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. |