Problem Analysis #4

GPT-oss-20B (high) · Run 1

110.03s
14,295 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

<|channel|>commentary to=python code<|message|>import mpmath as mp

def integral_n(n): f = lambda x: (1 + 205x/(69n))(-69*n) * mp.e(-6*x/7) return mp.quad(f, [0, n]) for n in [10, 20, 50, 100, 200]: val = integral_n(n) print(n, val)

Tool Output
[PYTHON OUTPUT] 10 0.0048647293405047951875408407345928852207051122885165 20 0.0048612310184988405879868706367926345285251358289216 50 0.004859134419088656446099320016853314804983205028148 100 0.0048584359510810701430545358252193303451270327675896 200 0.0048580867917418869602485710012266092670123367575959 [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>from fractions import Fraction Fraction(7,1441) float(Fraction(7,1441))

Tool Output
[PYTHON OUTPUT] 0.004857737682165163 [/PYTHON OUTPUT]
Reasoning Process

<|channel|>commentary to=python code<|message|>p = 7 q = 1441 print(p**2 * q)

Tool Output
[PYTHON OUTPUT] 70609 None [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] We have the integral

In=∫0n(1+20569nx)−69ne−67x dx.I_n=\int_{0}^{n}\left(1+\frac{205}{69n}x\right)^{-69n}e^{-\frac67 x}\,dx.

For large nn we can use the limit

lim⁡n→∞(1+an)bn=eab,\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^{bn}=e^{ab},

so

(1+20569nx)−69n=exp⁡ ⁣(−69n ln⁡(1+205x69n))⟶e−205x.\left(1+\frac{205}{69n}x\right)^{-69n} =\exp\!\bigl(-69n\,\ln(1+\tfrac{205x}{69n})\bigr) \longrightarrow e^{-205x}.

Thus the integrand tends to

e−205x e−67x=e−(205+67)x=e−14417x.e^{-205x}\,e^{-\frac67 x}=e^{-(205+\tfrac67)x} =e^{-\frac{1441}{7}x}.

The upper limit nn goes to ∞\infty, so the limiting integral is

∫0∞e−14417x dx=[ −71441e−14417x ]0∞=71441.\int_{0}^{\infty}e^{-\frac{1441}{7}x}\,dx =\left[\,-\frac{7}{1441}e^{-\frac{1441}{7}x}\,\right]_{0}^{\infty} =\frac{7}{1441}.

Hence p=7,  q=1441p=7,\;q=1441 (coprime). We need p2q=72×1441=49×1441=70609p^{2}q = 7^{2}\times1441=49\times1441=70609. Thus the answer is 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