| deepseek-ai/deepseek-math-7b-base |
| alignment-handbook |
| generated_from_trainer |
| aimo |
|
| example_title |
messages |
output |
| Math problem |
| role |
content |
| user |
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? |
|
|
| 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 |
|
|
|
text-generation |
| name |
results |
| NuminaMath-7B-TIR-GPTQ |
|
|
|
apache-2.0 |
|
