Model: AI-MO/NuminaMath-72B-TIR
Author: AI-MO
Pipeline: text-generation
Library: transformers
Tags: transformers, safetensors, qwen2, text-generation, alignment-handbook, generated_from_trainer, math, aimo, conversational, en, dataset:AI-MO/NuminaMath-TIR, base_model:AI-MO/NuminaMath-72B-CoT, base_model:finetune:AI-MO/NuminaMath-72B-CoT, license:other, text-generation-inference, endpoints_compatible, region:us
Base models (finetune): id: AI-MO/NuminaMath-72B-CoT
Card metadata: base_model: AI-MO/NuminaMath-72B-CoT
tags: alignment-handbook, generated_from_trainer, math, aimo
widget: example_title: Math problem
messages: user: 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 \\).


2- 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.


3- 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 \\).


4- From the equation \\( r_1 + r_2 = -k \\), for each valid pair \\((r_1, r_2)\\), we can compute the corresponding value of \\( k \\).


5- Finally, since we need the polynomial to have two distinct integer roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\) are distinct.
Let'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.
```python import itertools
# Find all pairs (r1, r2) such that r1 * r2 = 36 product_36 = 36 factor_pairs = []
for i in range(1, product_36 + 1):
  if product_36 % i == 0:
    pair = (i, product_36 // i)
    if pair[0] != pair[1]: # Ensure distinct pairs
      factor_pairs.append(pair)
    
    # Calculate k for each pair and ensure distinct integer roots
    valid_k_values = set()
    for r1, r2 in factor_pairs:
      if r1 != r2:
        k = -(r1 + r2)
        valid_k_values.add(k)
        
    print((len(valid_k_values), sorted(valid_k_values)))
 ```
 
 ```output
 (4, [-37, -20, -15,-13])
 ```
 The 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\\).
Therefore, the number of such values of \\( k \\) is:
[ \\boxed{4} \\]
pipeline_tag: text-generation
model-index: name: NuminaMath-72B-TIR
license: other
license_name: tongyi-qianwen
datasets: AI-MO/NuminaMath-TIR
language: en
Likes: 9 | All-time downloads: 9416
Created: 2024-07-17 06:58:44