Problem Analysis #20

DeepSeek V4 Pro · Run 1

Correct
17.17s
1,407 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(θ)=sin⁡θ−θ+θ36f(\theta)=\sin\theta-\theta+\frac{\theta^3}{6}

on −1≤θ≤1-1\le\theta\le1, let MM be the maximum value and mm be the minimum value. The individual values contain sin⁡1\sin1, so instead compute the integer

M+m.M+m.
Ground Truth
0
Recorded model response / solution
Reasoning Process

The function f(θ)=sin⁡θ−θ+θ36f(\theta) = \sin\theta - \theta + \frac{\theta^3}{6} is odd. Its derivative is f′(θ)=cos⁡θ−1+θ22f'(\theta) = \cos\theta - 1 + \frac{\theta^2}{2}. For θ>0\theta > 0, f′(θ)>0f'(\theta) > 0, and since f′f' is even, f′(θ)≥0f'(\theta) \ge 0 on [−1,1][-1,1] with equality only at 00. Thus ff is strictly increasing on [−1,1][-1,1]. The maximum occurs at θ=1\theta = 1 and the minimum at θ=−1\theta = -1.

M=f(1)=sin⁡1−1+16=sin⁡1−56M = f(1) = \sin 1 - 1 + \frac{1}{6} = \sin 1 - \frac{5}{6}
m=f(−1)=−sin⁡1+1−16=−sin⁡1+56m = f(-1) = -\sin 1 + 1 - \frac{1}{6} = -\sin 1 + \frac{5}{6}

Therefore, M+m=0M + m = 0.

\boxed{0}

Final Model Output
0

Comparative Analysis (P20) (Average across runs)

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

Avg Time Taken

K-EXAONE-236B-A23B
61.7
DeepSeek V4 Pro
24.7
Solar Pro 3
10.6
GPT-5.5
9.9
Gemini 3.5 Flash
7.9
Claude Opus 4.8
2.4
KT Mi:dm 2.0 Base Instruct
2.1

Avg Token Usage

K-EXAONE-236B-A23B
6031.7
Solar Pro 3
1824.3
Gemini 3.5 Flash
1761.0
KT Mi:dm 2.0 Base Instruct
1301.0
DeepSeek V4 Pro
1147.3
GPT-5.5
611.0
Claude Opus 4.8
417.3
    Tokyo · Problem 20 · DeepSeek V4 Pro | EntropyMath