File size: 6,595 Bytes
167ee6b f66744c 167ee6b f66744c 167ee6b f66744c 167ee6b f49478d 7027b1c f66744c 167ee6b d3eacdc f66744c 167ee6b f66744c 167ee6b f66744c 167ee6b f66744c f668db3 f66744c f668db3 f66744c f668db3 f66744c f668db3 f66744c d3eacdc 167ee6b f66744c d3eacdc f66744c |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 |
---
base_model: deepseek-ai/deepseek-math-7b-base
tags:
- alignment-handbook
- generated_from_trainer
datasets:
- AI-MO/tora-chosen-v0.7
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://cdn-uploads.huggingface.co/production/uploads/6200d0a443eb0913fa2df7cc/4xNbaeRi6HaAeo7UoRDZR.png" alt="Zephyr Logo" width="800" style="margin-left:'auto' margin-right:'auto' display:'block'"/>
# Model Card for Numina-Math-7B
Numina-Math is a series of language models that are trained to solve math problems using tool integrated reasoning.
Numina-Math-7b won the first AI Math Olympiad, 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 on Math Question answers and multi-step synthetic generations using tool integrated reasoning.
## Model description
- **Model type:** A 7B parameter Math model fine-tuned in two stages of training, first on a dataset with 863k math question answer pairs and then on a dataset with 73k examples of multi-step synthetic generations using tool integrated reasoning.
- **Language(s) (NLP):** Primarily English
- **License:** MIT
- **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
- **Demo:** https://huggingface.co/spaces/AI-MO/math-olympiad-solver
## 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"],
"tokenizer": pipe.tokenizer,
}
outputs = pipe(prompt, **gen_config)
text = outputs[0]["generated_text"]
print(text)
python_code = re.findall(r"```python(.*?)```", text, re.DOTALL)[0]
# 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.
exec(python_code)
```
## Bias, Risks, and Limitations
<!-- This section is meant to convey both technical and sociotechnical limitations. -->
Numina-Math-7B was created to solve math problems, the model has not been aligned to preferences beyond the domain of solving math, and should not be used in a general chat setting.
## 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
## Citation
If you find Numina-Math useful in your work, please cite it with:
```
@misc{beeching2024numina-math,
title={Numina Math},
author={Edward Beeching and Lewis Tunstall and Roman Soletskyi and Kashif Rasul and Shengyi Huang and Jia Li},
year={2024},
publisher = {Hugging Face},
journal = {Hugging Face repository},
howpublished = {\url{https://huggingface.co/AI-MO/Numina-Math-7B}}
}
``` |