cs-552-2026-centralesupechec/math_model

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ModelHub XC c3e312fc28 初始化项目，由ModelHub XC社区提供模型

Model: cs-552-2026-centralesupechec/math_model
Source: Original Platform

2026-06-13 10:10:18 +08:00

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Automated MNLP evaluation report

Model repo: cs-552-2026-centralesupechec/math_model
Owner(s): group centralesupechec
Generated at: 2026-05-13T04:17:52+00:00 (UTC)
Pipeline: mnlp-project-ci

This PR is opened automatically by the course CI. It is non-blocking — you do not need to merge it. The next nightly run will refresh this file.

Summary

Benchmark	Accuracy	Status
Math	0.1700	ok
Knowledge	—	not run
Multilingual	—	not run
Safety	—	not run

Sample completions

Prompts are intentionally omitted to avoid revealing benchmark contents. For multi-completion problems, only one completion is shown per sample.

Math

Correct (1 shown)

reference: 2
overall (3/8 completions correct)
extracted (✓): 2

completion:

<think>
Okay, so I need to solve the equation $3^{2x} + 19 = 10^x$. Let me think about how to approach this. 

First, let me rewrite the equation to see if I can simplify it or recognize any patterns. The left side has $3^{2x}$, which is the same as $(3^2)^x = 9^x$. So, the equation becomes $9^x + 19 = 10^x$. 

Hmm, so now it's $9^x + 19 = 10^x$. I don't see an obvious way to combine these terms,

... [truncated 821 chars] ...

9^x + 19 - 10^x
$$

We seek the value(s) of $x$ for which $f(x) = 0$.

---

### Step 2: Test Integer Values

Let’s test some integer values of $x$:

- **At $x = 2$:**

$$
f(2) = 9^2 + 19 - 10^2 = 81 + 19 - 100 = 100 - 100 = 0
$$

So, $x = 2$ is a solution.

---

### Step 3: Analyze the Behavior of the Function

Define the function:

$$
f(x) = 9^x + 19 - 10^x
$$

As $x \to \infty$, the exponential term $10^x$ dominates, so $f(x) \to -\infty$.

As $x \to -\infty$, both $9^x$ and $10^x$ approach 0, so $f(x) \to 19$.

Now, compute the derivative:

$$
f'(x) = \ln(9) \cdot 9^x - \ln(10) \cdot 10^x
$$

This derivative is always negative for all $x$, since:

- $\ln(9) \approx 2.1972$
- $\ln(10) \approx 2.3026$
- $9^x$ grows slower than $10^x$, and the coefficient of $9^x$ is smaller than that of $10^x$

Thus, $f'(x) < 0$ for all $x$, meaning the function is strictly decreasing.

---

### Step 4: Conclude the Number of Solutions

Since $f(x)$ is strictly decreasing and continuous, and it goes from $19$ at $-\infty$ to $-\infty$ at $+\infty$, it can cross the x-axis **exactly once**.

We already found that $f(2) = 0$, so this is the **only** solution.

---

### Final Answer

$$
\boxed{2}
$$

Incorrect (1 shown)

reference: $(n-2)2^n +1$
overall (0/8 completions correct)
extracted (✗): <no answer>