Problem Analysis #11

K-EXAONE-236B-A23B · Run 1

Correct
97.73s
13,282 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.

In an isosceles triangle ABCABC with AB=AC=1AB=AC=1, let ss be the smallest radius such that the three disks centered at A,B,CA,B,C cover all three sides. For ∠BAC=π/3\angle BAC=\pi/3, s=1/2s=1/2. For ∠BAC=2π/3\angle BAC=2\pi/3, s=3/2s=\sqrt3/2. Compute the integer

(2s1)+(4s22),(2s_1)+(4s_2^2),

where s1s_1 is the first value and s2s_2 is the second value.

Ground Truth
4
Recorded model response / solution
Reasoning Process

We are given an isosceles triangle ABCABC with AB=AC=1AB = AC = 1, and for two specific angles at vertex AA, we are told the value of the smallest radius ss such that the three disks centered at A,B,CA, B, C cover all three sides of the triangle.

Specifically:

  • For \angle BAC = $$\frac{\pi}{3}$$, it is stated that s_1 = $$\frac{1}{2}$$
  • For \angle BAC = $$\frac{2\pi}{3}$$, it is stated that s_2 = $$\frac{\sqrt{3}}{2}$$

We are asked to compute the integer:

(2s1)+(4s22)(2s_1) + (4s_2^2)

Step 1: Compute 2s12s_1

2s1=2⋅12=12s_1 = 2 \cdot \frac{1}{2} = 1

Step 2: Compute 4s224s_2^2

s22=(32)2=34s_2^2 = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} 4s22=4⋅34=34s_2^2 = 4 \cdot \frac{3}{4} = 3

Step 3: Add the two results

(2s1)+(4s22)=1+3=4(2s_1) + (4s_2^2) = 1 + 3 = 4

Final Answer

4\boxed{4}
Final Model Output
4

Comparative Analysis (P11) (Average across runs)

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

Avg Time Taken

K-EXAONE-236B-A23B
132.8
Gemini 3.5 Flash
18.0
DeepSeek V4 Pro
7.6
GPT-5.5
3.5
Solar Pro 3
2.8
Claude Opus 4.8
2.5
KT Mi:dm 2.0 Base Instruct
1.3

Avg Token Usage

K-EXAONE-236B-A23B
17931.0
Gemini 3.5 Flash
4173.7
KT Mi:dm 2.0 Base Instruct
1188.3
Solar Pro 3
645.3
DeepSeek V4 Pro
578.7
Claude Opus 4.8
413.3
GPT-5.5
351.0
    Tokyo · Problem 11 · K-EXAONE-236B-A23B | EntropyMath