Problem Analysis #4

GPT-5.1 (high) · Run 1

25.75s
3,162 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] I will compute the expected limit by recognizing the integrand limit and then confirm the final arithmetic with Python. [/THOUGHT] [PYTHON]

Compute p^2 * q where the expected limit is 7/1441

p = 7 q = 1441 print(p**2 * q) [/PYTHON]

Tool Output
[PYTHON OUTPUT] 70609 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] For each fixed xx, as n→∞n\to\infty,

(1+20569nx)−69n→e−205x.\left(1+\frac{205}{69n}x\right)^{-69n} \to e^{-205x}.

Thus the integrand converges pointwise to

e−205xe−67x=e−(205+67)x=e−14417x.e^{-205x} e^{-\frac{6}{7}x} = e^{-\left(205+\frac{6}{7}\right)x} = e^{-\frac{1441}{7}x}.

One can dominate the integrand by an exponentially decaying function independent of nn, so by dominated convergence the limit of the integrals equals the integral of the limit:

lim⁡n→∞∫0n(1+20569nx)−69ne−67x dx=∫0∞e−14417x dx=114417=71441.\lim_{n\to\infty} \int_0^n \left(1+\frac{205}{69n}x\right)^{-69n} e^{-\frac{6}{7}x}\,dx = \int_0^\infty e^{-\frac{1441}{7}x}\,dx = \frac{1}{\frac{1441}{7}} = \frac{7}{1441}.

Thus p=7p=7, q=1441q=1441, so

p2q=72⋅1441=70609.p^2 q = 7^2\cdot 1441 = 70609.

[/THOUGHT] 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