Problem Analysis #20

Gemini 3.5 Flash · Run 1

Correct
7.24s
1,447 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

To find the sum of the maximum value MM and the minimum value mm of the function

f(θ)=sin⁡θ−θ+θ36f(\theta)=\sin\theta-\theta+\frac{\theta^3}{6}

on the interval [−1,1][-1, 1], we first observe the symmetry of the function.

We test if f(θ)f(\theta) is an odd function:

f(−θ)=sin⁡(−θ)−(−θ)+(−θ)36=−sin⁡θ+θ−θ36=−f(θ)f(-\theta) = \sin(-\theta) - (-\theta) + \frac{(-\theta)^3}{6} = -\sin\theta + \theta - \frac{\theta^3}{6} = -f(\theta)

Since f(−θ)=−f(θ)f(-\theta) = -f(\theta), the function f(θ)f(\theta) is odd.

For any odd function defined on a symmetric interval [−a,a][-a, a], the maximum value MM and the minimum value mm are negatives of each other. That is, if the maximum value MM occurs at some point θ0∈[−1,1]\theta_0 \in [-1, 1], then the minimum value mm must occur at −θ0-\theta_0, giving:

m=f(−θ0)=−f(θ0)=−Mm = f(-\theta_0) = -f(\theta_0) = -M

Thus, the sum of the maximum and minimum values is:

M+m=M+(−M)=0M + m = M + (-M) = 0

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
6,032
Solar Pro 3
1,824
Gemini 3.5 Flash
1,761
KT Mi:dm 2.0 Base Instruct
1,301
DeepSeek V4 Pro
1,147
GPT-5.5
611
Claude Opus 4.8
417
    Tokyo · Problem 20 · Gemini 3.5 Flash | EntropyMath