metadata
base_model: AI-MO/NuminaMath-72B-CoT
tags:
- alignment-handbook
- generated_from_trainer
- math
- 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 \\).
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
results: []
license: other
license_name: tongyi-qianwen
datasets:
- AI-MO/NuminaMath-TIR
language:
- en
Model Card for NuminaMath 72B TIR
NuminaMath is a series of language models that are trained with two stages of supervised fine-tuning to solve math problems using chain of thought (CoT) and tool-integrated reasoning (TIR):
- Stage 1: fine-tune the base model on a large, diverse dataset of natural language math problems and solutions, where each solution is templated with Chain of Thought (CoT) to facilitate reasoning.
- Stage 2: fine-tune the model from Stage 1 on a synthetic dataset of tool-integrated reasoning, where each math problem is decomposed into a sequence of rationales, Python programs, and their outputs.
Model description
- Model type: A 72B parameter math LLM fine-tuned on a dataset with 860k+ math problem-solution pairs.
- Language(s) (NLP): Primarily English
- License: Tongyi Qianwen
- Finetuned from model: Qwen/Qwen2-72B
Model Sources
Intended uses & limitations
Here's how you can run the model using the pipeline() function from 🤗 Transformers:
import re
import torch
from transformers import pipeline
pipe = pipeline("text-generation", model="AI-MO/NuminaMath-72B-TIR", torch_dtype=torch.bfloat16, device_map="auto")
messages = [
{"role": "user", "content": "For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?"},
]
prompt = pipe.tokenizer.apply_chat_template(messages, tokenize=False, add_generation_prompt=True)
gen_config = {
"max_new_tokens": 1024,
"do_sample": False,
"stop_strings": ["```output"], # Generate until Python code block is complete
"tokenizer": pipe.tokenizer,
}
outputs = pipe(prompt, **gen_config)
text = outputs[0]["generated_text"]
print(text)
# WARNING: This code will execute the python code in the string. We show this for eductional purposes only.
# Please refer to our full pipeline for a safer way to execute code.
python_code = re.findall(r"```python(.*?)```", text, re.DOTALL)[0]
exec(python_code)
The above executes a single step of Python code - for more complex problems, you will want to run the logic for several steps to obtain the final solution.
Bias, Risks, and Limitations
NuminaMath 72B TIR was created to solve problems in the narrow domain of competition-level mathematics. As a result, the model should not be used for general chat applications. With greedy decoding, we find the model is capable of solving problems at the level of AMC 12, but often struggles generate a valid solution on harder problems at the AIME and Math Olympiad level. The model also struggles to solve geometry problems, likely due to it's limited capacity and lack of other modalities like vision.
Training procedure
Training hyperparameters
The following hyperparameters were used during training:
- learning_rate: 2e-05
- train_batch_size: 1
- eval_batch_size: 1
- seed: 42
- distributed_type: multi-GPU
- num_devices: 32
- total_train_batch_size: 32
- total_eval_batch_size: 32
- optimizer: Adam with betas=(0.9,0.999) and epsilon=1e-08
- lr_scheduler_type: cosine
- lr_scheduler_warmup_ratio: 0.1
- num_epochs: 4
Framework versions
- Transformers 4.42.3
- Pytorch 2.3.0+cu121
- Datasets 2.18.0
- Tokenizers 0.19.1
Citation
If you find NuminaMath 7B TIR is useful in your work, please cite it with:
@misc{numina_math_7b,
author = {Edward Beeching and Shengyi Costa Huang and Albert Jiang and Jia Li and Benjamin Lipkin and Zihan Qina and Kashif Rasul and Ziju Shen and Roman Soletskyi and Lewis Tunstall},
title = {NuminaMath 7B TIR},
year = {2024},
publisher = {Numina & Hugging Face},
journal = {Hugging Face repository},
howpublished = {\url{https://huggingface.co/AI-MO/NuminaMath-7B-TIR}}
}