Reasoning模型蒸馏实践：用大模型提升小模型能力

Okay, so I need to solve this system of equations:x + y + z = 3,x² + y² + z² = 3,x³ + y³ + z³ = 3.Hmm, let's see. These are symmetric equations, so maybe I can use some symmetric properties or formulas. Let me recall that for symmetric equations, using variables like sums and products might help. Let me denote S = x + y + z, which is given as 3. Then, the sum of squares is given by x² + y² + z² = 3. I also know that the sum of squares can be related to S and the pairwise products. The formula is:x² + y² + z² = (x + y + z)² - 2(xy + yz + zx).Plugging in the known values, 3 = (3)² - 2(xy + yz + zx). So, 3 = 9 - 2(xy + yz + zx). Let's solve for the sum of products:2(xy + yz + zx) = 9 - 3 = 6 ⇒ xy + yz + zx = 3.Okay, so now I have S = 3, and the sum of products is also 3. Now, moving on to the third equation, which is the sum of cubes: x³ + y³ + z³ = 3. There's a formula for the sum of cubes in terms of S, the sum of products, and the product xyz. The formula is:x³ + y³ + z³ = (x + y + z)³ - 3(x + y + z)(xy + yz + zx) + 3xyz.Plugging in the known values:3 = 3³ - 3*3*3 + 3xyz.Calculating each term:3³ is 27,3*3*3 is 27,So,3 = 27 - 27 + 3xyz ⇒ 3 = 0 + 3xyz ⇒ 3xyz = 3 ⇒ xyz = 1.So now I have S = 3, sum of products = 3, and product = 1. These are the elementary symmetric sums. Therefore, the monic cubic polynomial whose roots are x, y, z is:t³ - S t² + (xy + yz + zx) t - xyz = 0 ⇒ t³ - 3t² + 3t - 1 = 0.So, solving this cubic equation should give me all real numbers x, y, z. Let me try to factor this cubic. Maybe it has rational roots? By Rational Root Theorem, possible roots are ±1. Let's test t=1:1³ - 3*1² + 3*1 -1 = 1 -3 +3 -1 = 0. Yes, t=1 is a root. So, we can factor out (t -1). Let's perform polynomial division or use synthetic division.Dividing t³ -3t² +3t -1 by (t -1):Using synthetic division:1 | 1  -3   3   -1           1  -2    1       1  -2    1    0So, the quadratic factor is t² - 2t +1. Which factors further as (t -1)².Therefore, the cubic factors as (t -1)³ = 0. So, all three roots are 1. Therefore, the solution is x = y = z =1.Wait, but let me check if there could be other solutions. Suppose the cubic factors into (t-1)^3, which means that all three variables are 1. Let me verify if this satisfies the original equations.First equation: 1 +1 +1 =3, yes.Second: 1² +1² +1² =3, yes.Third:1³ +1³ +1³=3, yes. So that works.But wait, maybe there are other real solutions where variables are not all 1? Let's see. Since the cubic equation only has t=1 as a triple root, that's the only solution. Therefore, the only real solution is x=y=z=1.But let me think again. Suppose that two variables are equal to 1 and the third is different. Let me test that. Suppose x =1, y=1, then z =3 -1 -1=1. So, z=1. So that's still all ones. Alternatively, if one variable is not 1, would that work?Wait, suppose that x, y, z are all 1, which works. Suppose, for example, x=2, y=0, z=1. Then x+y+z=3. Check the sum of squares: 4 +0 +1=5 ≠3. So that doesn't work. Alternatively, maybe some other combination. But given that the cubic equation only has 1 as a root, multiplicity three, that suggests that all three variables must be 1.Alternatively, perhaps complex roots, but the problem asks for real numbers. So, since the only real root is 1 with multiplicity three, the only real solution is x=y=z=1. Therefore, the answer is all variables equal to 1.<answer>The only real solution is \( x = y = z = 1 \). Thus, the numbers are \(\boxed{1}\), \(\boxed{1}\), and \(\boxed{1}\).Alternatively, since all variables are the same, it can be written as \(\boxed{1}\) for each variable.**Final Answer**The real numbers are \(\boxed{1}\), \(\boxed{1}\), and \(\boxed{1}\).</answer>

[  {    "role": "user",    "content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad  \\textbf{(B)}\\ 7 \\qquad  \\textbf{(C)}\\ 8 \\qquad  \\textbf{(D)}\\  9\\qquad  \\textbf{(E)}\\ \\text{cannot be determined}$\nPlease reason step by step, and put your final answer within \boxed{}."  },  {    "role": "assistant",    "content": "\nOkay, let's see. I need to figure out which ....... Answer is B.\n\n**Final Answer**\n\\boxed{B}\n\n\nGiven a set of consecutive positive integers starting ...... Average of remaining numbers: \\(\\frac{2408}{68} = \\frac{602}{17} = 35 \\frac{7}{17}\\)\n\nThus, the number erased is \\(\\boxed{B}\\)."  }]

!CUDA_VISIBLE_DEVICES=0 \ swift sft \    --model Qwen/Qwen2.5-1.5B-Instruct \    --train_type lora \    --lora_rank 16 \    --torch_dtype bfloat16 \    --dataset 'modelscope/MathR:clean' \    --split_dataset_ratio 0 \    --max_length 4096 \    --num_train_epochs 1 \    --per_device_train_batch_size 1 \    --learning_rate 1e-5 \    --gradient_accumulation_steps 16 \    --save_steps 100 \    --save_total_limit 10 \    --logging_steps 5 \    --report_to swanlab \    --swanlab_token YOUR_SWANLAB_TOKEN \    --swanlab_mode cloud

swift infer \--adapters 'output/Qwen2.5-1.5B-Instruct/v11-20250415-120200/checkpoint-81' \--stream true \--infer_backend pt \--temperature 0 \--max_new_tokens 2048

from evalscope import run_task, TaskConfigtask_config = TaskConfig(    model="Qwen/Qwen2.5-1.5B-Instruct",  # 原始模型    datasets=["gsm8k"],  # 数据集名称    dataset_args={        "gsm8k": {"few_shot_num": 0},  # few_shot_num: 0表示不使用few-shot    },    generation_config={        "max_new_tokens": 4096,  # 生成的最大token数        "temperature": 0,  # 生成的温度系数，0表示贪婪搜索    },    eval_batch_size=10,  # 评测时的batch size    limit=100   # 评测数据集的大小，抽取前100条数据进行评测)run_task(task_config)

!swift export \    --adapters /mnt/data/data/user/maoyunlin.myl/tools/course/distill/output/Qwen2.5-1.5B-Instruct/v11-20250415-120200/checkpoint-81 \    --merge_lora true

# 测试蒸馏训练后的模型from evalscope import run_task, TaskConfig# 记得替换下面的model路径task_config = TaskConfig(    model="/mnt/data/data/user/maoyunlin.myl/tools/course/distill/output/Qwen2.5-1.5B-Instruct/v11-20250415-120200/checkpoint-81-merged",    datasets=["gsm8k"],    dataset_args={        "gsm8k": {"few_shot_num": 0},    },    generation_config={        "max_new_tokens": 4096,        "temperature": 0,    },    eval_batch_size=10,    limit=100)run_task(task_config)

import osos.environ['GRADIO_ROOT_PATH'] = f"/{os.environ['JUPYTER_NAME']}/proxy/7860"print(os.environ['GRADIO_ROOT_PATH'])

!evalscope app