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---
base_model: deepseek-ai/deepseek-math-7b-base
tags:
- alignment-handbook
- generated_from_trainer
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: Numina-Math-7B
results: []
---
<!-- 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/Numina-Math-7B/resolve/main/thumbnail.png" alt="Numina Logo" width="800" style="margin-left:'auto' margin-right:'auto' display:'block'"/>
# Model Card for NuminaMath 7B
NuminaMath is a series of language models that are trained to solve math problems using tool-integrated reasoning. NuminaMath 7B 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 a fine-tuned version of [deepseek-ai/deepseek-math-7b-base](https://huggingface.co/deepseek-ai/deepseek-math-7b-base) with two stages of supervised fine-tuning:
* **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 learning.
* **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. Here we followed [Microsoft’s ToRA paper](https://arxiv.org/abs/2309.17452) and prompted GPT-4 to produce solutions in the ToRA format with code execution feedback. Fine-tuning on this data produces a reasoning agent that can solve mathematical problems via a mix of natural language reasoning and use of the Python REPL to compute intermediate results.
## Model description
- **Model type:** A 7B parameter math LLM fine-tuned in two stages of supervised fine-tuning, first on a dataset with math problem-solution pairs and then on a synthetic dataset with examples of multi-step generations using tool-integrated reasoning.
- **Language(s) (NLP):** Primarily English
- **License:** Apache 2.0
- **Finetuned from model:** [deepseek-ai/deepseek-math-7b-base](https://huggingface.co/deepseek-ai/deepseek-math-7b-base)
### Model Sources
<!-- Provide the basic links for the model. -->
- **Repository:** Coming soon to https://github.com/huggingface/alignment-handbook
## Intended uses & limitations
Here's how you can run the model using the `pipeline()` function from 🤗 Transformers:
```python
import re
import torch
from transformers import pipeline
pipe = pipeline("text-generation", model="AI-MO/Numina-Math-7B", 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
<!-- This section is meant to convey both technical and sociotechnical limitations. -->
NuminaMath 7B 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](https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems), but often struggles generate a valid solution on harder problems at the AIME and Math Olympiad level.
## Training procedure
### Training hyperparameters
The following hyperparameters were used during training:
- learning_rate: 2e-05
- train_batch_size: 4
- eval_batch_size: 8
- seed: 42
- distributed_type: multi-GPU
- num_devices: 8
- total_train_batch_size: 32
- total_eval_batch_size: 64
- optimizer: Adam with betas=(0.9,0.999) and epsilon=1e-08
- lr_scheduler_type: cosine
- lr_scheduler_warmup_ratio: 0.1
- num_epochs: 4.0
### Training results
| Training Loss | Epoch | Step | Validation Loss |
|:-------------:|:-----:|:----:|:---------------:|
| 0.4295 | 1.0 | 1733 | 0.4313 |
| 0.3638 | 2.0 | 3466 | 0.4332 |
| 0.2951 | 3.0 | 5199 | 0.4704 |
| 0.2225 | 4.0 | 6932 | 0.5302 |
### Framework versions
- Transformers 4.40.1
- Pytorch 2.3.1
- Datasets 2.18.0
- Tokenizers 0.19.1 |