Problem Analysis #4

Gemma-3-27B · Run 1

32.96s
1,741 toks
Problem Statement

Suppose the following limit is given:

lim⁡n→∞∫[0,n](1+20569nx)−69ne(−67x) dx=pq,\lim_{n\to\infty} \int_{[0,n]} \left(1+\frac{205}{69n}x\right)^{-69n} e^{(-\frac{6}{7}x)}\,dx=\frac{p}{q},

where pp and qq are positive integers which are relatively prime. Find p2qp^2q.

\noindent

Ground Truth
70609
Execution Trace
Reasoning Process

[THOUGHT] Let f(x)=(1+20569nx)−69ne(−67x)f(x) = \left(1+\frac{205}{69n}x\right)^{-69n} e^{(-\frac{6}{7}x)}. As n→∞n \to \infty, we have 20569n→0\frac{205}{69n} \to 0. Thus,

(1+20569nx)−69n≈e−20569nx⋅69n=e−205x.\left(1+\frac{205}{69n}x\right)^{-69n} \approx e^{-\frac{205}{69n}x \cdot 69n} = e^{-205x}.

Therefore, the integral becomes

lim⁡n→∞∫[0,n]e−205x−67x dx=lim⁡n→∞∫0ne−(205+67)x dx.\lim_{n\to\infty} \int_{[0,n]} e^{-205x-\frac{6}{7}x}\,dx = \lim_{n\to\infty} \int_0^n e^{-(205+\frac{6}{7})x}\,dx.

We have 205+67=1435+67=14417205 + \frac{6}{7} = \frac{1435+6}{7} = \frac{1441}{7}.

lim⁡n→∞∫0ne−14417x dx=lim⁡n→∞[−71441e−14417x]0n=lim⁡n→∞−71441e−14417n−(−71441e0)=lim⁡n→∞−71441e−14417n+71441.\lim_{n\to\infty} \int_0^n e^{-\frac{1441}{7}x}\,dx = \lim_{n\to\infty} \left[-\frac{7}{1441}e^{-\frac{1441}{7}x}\right]_0^n = \lim_{n\to\infty} -\frac{7}{1441}e^{-\frac{1441}{7}n} - \left(-\frac{7}{1441}e^0\right) = \lim_{n\to\infty} -\frac{7}{1441}e^{-\frac{1441}{7}n} + \frac{7}{1441}.

Since lim⁡n→∞e−14417n=0\lim_{n\to\infty} e^{-\frac{1441}{7}n} = 0, we have

lim⁡n→∞∫0ne−(205+67)x dx=71441.\lim_{n\to\infty} \int_0^n e^{-(205+\frac{6}{7})x}\,dx = \frac{7}{1441}.

Thus, p=7p=7 and q=1441q=1441. We need to check if they are relatively prime. 1441=7⋅205+61441 = 7 \cdot 205 + 6. Thus gcd⁡(7,1441)=gcd⁡(7,6)=1\gcd(7,1441) = \gcd(7,6)=1. Then p2q=72⋅1441=49⋅1441=70609p^2q = 7^2 \cdot 1441 = 49 \cdot 1441 = 70609.

p = 7
q = 1441
print(p**2 * q)
Tool Output
[PYTHON OUTPUT] 70609 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python output confirms the calculation 72⋅1441=706097^2 \cdot 1441 = 70609.

70609\boxed{70609}

Final Model Output
70609

Comparative Analysis (P4) (Average across runs)

Avg Time Taken

Solar-Open-100B
438.0
Deepseek-R1-distill-Qwen-32B (high)
140.9
Deepseek-V3.2
123.2
Gemini-3-Pro-Preview
90.8
Kanana-2-30B-Thinking
87.7
GPT-oss-20B (high)
60.4
Solar-Pro-2 (31B)(high)
60.2
K-EXAONE-236B-A23B
53.2
Grok-4.1-fast
38.4
HCX-007(high)
31.5
Gemma-3-27B
31.4
Qwen3-30B-A3B-2507
28.9
Kanana-2-30B-Thinking-2601
22.7
Llama-VARCO-8B-Instruct
21.3
GPT-5.1 (high)
19.0
Claude-Opus-4.5
16.4
A.X-4.0 (72B)
16.0
EXAONE-4.0.1-32B (high)
3.7

Avg Token Usage

K-EXAONE-236B-A23B
20,406
Kanana-2-30B-Thinking
13,425
Gemini-3-Pro-Preview
11,241
Deepseek-V3.2
9,875
Solar-Pro-2 (31B)(high)
9,831
GPT-oss-20B (high)
9,757
Solar-Open-100B
8,096
Grok-4.1-fast
7,672
Kanana-2-30B-Thinking-2601
6,314
Qwen3-30B-A3B-2507
6,267
HCX-007(high)
4,190
Deepseek-R1-distill-Qwen-32B (high)
3,984
Claude-Opus-4.5
3,858
A.X-4.0 (72B)
3,485
EXAONE-4.0.1-32B (high)
3,238
GPT-5.1 (high)
3,068
Llama-VARCO-8B-Instruct
2,088
Gemma-3-27B
1,706