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---
base_model: deepseek-ai/deepseek-math-7b-base
tags:
- alignment-handbook
- generated_from_trainer
- aimo
widget:
- example_title: Math problem
messages:
- role: user
content: >-
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$
have two distinct integer roots?
output:
text: >-
### Solution: 1. For the polynomial \\( x^2 + kx + 36 \\) to have two
distinct integer roots, let's denote these roots by \\( r_1 \\) and \\(
r_2 \\).\n\n2. According to Vieta's formulas, the sum of the roots \\( r_1
+ r_2 \\) is equal to \\(-k\\), and the product of the roots \\( r_1
\\cdot r_2 \\) is equal to 36.\n\n3. To find the distinct integer pairs
\\((r_1, r_2)\\) whose product is 36, we need to determine all pairs of
integers \\((r_1, r_2)\\) such that \\( r_1 \\cdot r_2 = 36 \\) and \\(
r_1 \\neq r_2 \\).\n\n4. From the equation \\( r_1 + r_2 = -k \\), for
each valid pair \\((r_1, r_2)\\), we can compute the corresponding value
of \\( k \\).\n\n5. Finally, since we need the polynomial to have two
distinct integer roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\)
are distinct.\n\nLet's start by finding all pairs \\((r_1, r_2)\\) such
that \\( r_1 \\cdot r_2 = 36 \\). We'll then determine the values of \\( k
\\) and ensure the roots are distinct.\n```python\nimport itertools\n\n#
Find all pairs (r1, r2) such that r1 * r2 = 36\nproduct_36 =
36\nfactor_pairs = []\n\nfor i in range(1, product_36 + 1):\n if
product_36 % i == 0:\n pair = (i, product_36 // i)\n if pair[0] !=
pair[1]: # Ensure distinct pairs\n factor_pairs.append(pair)\n\n#
Calculate k for each pair and ensure distinct integer
roots\nvalid_k_values = set()\nfor r1, r2 in factor_pairs:\n if r1 !=
r2:\n k = -(r1 + r2)\n
valid_k_values.add(k)\n\nprint((len(valid_k_values),
sorted(valid_k_values)))\n```\n```output\n(4, [-37, -20, -15,
-13])\n```\nThe distinct integer values of \\( k \\) that make the
polynomial \\( x^2 + kx + 36 \\) have two distinct integer roots are
\\(-37, -20, -15, \\text{and} -13\\).\n\nTherefore, the number of such
values of \\( k \\) is:\n\n\\[\n\\boxed{4}\n\\]\n
pipeline_tag: text-generation
model-index:
- name: NuminaMath-7B-TIR-GPTQ
results: []
license: apache-2.0
language:
- en
---
<!-- This model card has been generated automatically according to the information the Trainer had access to. You
should probably proofread and complete it, then remove this comment. -->
<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'"/>
# Model Card for NuminaMath 7B TIR GPTQ
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.
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. |