--- 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 ---

# 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. ![image/png](https://cdn-uploads.huggingface.co/production/uploads/6200d0a443eb0913fa2df7cc/NyhBs_gzg40iwL995DO9L.png) 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.