Problem Analysis #11

Gemini 3.5 Flash · Run 1

Correct

22.89s

4,979 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 $ABC$ with $AB=AC=1$ , let $s$ be the smallest radius such that the three disks centered at $A,B,C$ cover all three sides. For $\angle BAC=\pi/3$ , $s=1/2$ . For $\angle BAC=2\pi/3$ , $s=\sqrt3/2$ . Compute the integer

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

where $s_1$ is the first value and $s_2$ is the second value.

Ground Truth

Recorded model response / solution

Reasoning Process

To find the value of the expression $(2s_1) + (4s_2^2)$ , we use the given values for $s_1$ and $s_2$ :

The first value is $s_1 = \frac{1}{2}$ (for $\angle BAC = \pi/3$ ).
The second value is $s_2 = \frac{\sqrt{3}}{2}$ (for $\angle BAC = 2\pi/3$ ).

Now, we substitute these values into the expression:

Compute $2s_1$ :
$2s_1 = 2 \left(\frac{1}{2}\right) = 1$
Compute $4s_2^2$ :
$4s_2^2 = 4 \left(\frac{\sqrt{3}}{2}\right)^2 = 4 \left(\frac{3}{4}\right) = 3$
Add the two results:
$(2s_1) + (4s_2^2) = 1 + 3 = 4$

\boxed{4}

Final Model Output

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