Problem Analysis #5

Claude Opus 4.8 · Run 1

Correct
3.51s
616 toks

API · no tools. This is the prompt and response recorded for this attempt; a full solution may not have been returned.

Problem Statement

Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.

For

f(x)=∫01∣t−x∣1+t2 dt(0≤x≤1),f(x)=\int_0^1\frac{|t-x|}{1+t^2}\,dt\qquad(0\le x\le1),

let α\alpha be the number satisfying 0<α<π/40<\alpha<\pi/4 and f′(tan⁡α)=0f'(\tan\alpha)=0. Let u=tan⁡αu=\tan\alpha. The value of uu is irrational, so instead compute the integer

(u+1u)2.\left(u+\frac1u\right)^2.
Ground Truth
8
Recorded model response / solution
Reasoning Process

Let me compute f(x).

f(x) = ∫₀ˣ (x-t)/(1+t²) dt + ∫ₓ¹ (t-x)/(1+t²) dt

f'(x): differentiate. Using Leibniz, the boundary terms vanish (integrand zero at t=x).

f'(x) = ∫₀ˣ 1/(1+t²) dt - ∫ₓ¹ 1/(1+t²) dt = arctan(x) - (arctan(1) - arctan(x)) = 2arctan(x) - π/4.

Set to 0: arctan(x) = π/8, so x = tan(π/8) = √2 - 1.

So u = tan(π/8) = √2 - 1.

1/u = 1/(√2-1) = √2+1.

u + 1/u = 2√2. Squared: 8.

\boxed{8}

Final Model Output
8

Comparative Analysis (P5) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

K-EXAONE-236B-A23B
24.1
DeepSeek V4 Pro
18.2
Gemini 3.5 Flash
11.7
GPT-5.5
8.0
Solar Pro 3
6.9
Claude Opus 4.8
3.5
KT Mi:dm 2.0 Base Instruct
3.2

Avg Token Usage

K-EXAONE-236B-A23B
3597.7
Gemini 3.5 Flash
2820.0
KT Mi:dm 2.0 Base Instruct
1642.3
Solar Pro 3
1309.0
DeepSeek V4 Pro
1304.0
Claude Opus 4.8
590.0
GPT-5.5
559.3
    Tokyo · Problem 5 · Claude Opus 4.8 | EntropyMath