Problem Analysis #2

Gemini 3.5 Flash · Run 1

Correct
4.02s
864 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 the setting of the original midpoint-angle problem, the condition q=1/3q=1/3 leads to a positive value pp satisfying

p=3−22.p=3-2\sqrt2.

Instead of giving this irrational value, compute the integer

N=p+1p.N=p+\frac1p.
Ground Truth
6
Recorded model response / solution
Reasoning Process

To find the integer N=p+1pN = p + \frac{1}{p} where p=3−22p = 3 - 2\sqrt{2}, we first calculate the reciprocal of pp:

1p=13−22\frac{1}{p} = \frac{1}{3 - 2\sqrt{2}}

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate $3 + 2$$\sqrt{2}$$$:

1p=3+22(3−22)(3+22)=3+2232−(22)2=3+229−8=3+22\frac{1}{p} = \frac{3 + 2\sqrt{2}}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} = \frac{3 + 2\sqrt{2}}{3^2 - (2\sqrt{2})^2} = \frac{3 + 2\sqrt{2}}{9 - 8} = 3 + 2\sqrt{2}

Now, we sum pp and 1p\frac{1}{p}:

N=p+1p=(3−22)+(3+22)=6N = p + \frac{1}{p} = (3 - 2\sqrt{2}) + (3 + 2\sqrt{2}) = 6

The final answer is:

\boxed{6}

Final Model Output
6

Comparative Analysis (P2) (Average across runs)

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

Avg Time Taken

DeepSeek V4 Pro
10.2
K-EXAONE-236B-A23B
8.7
Gemini 3.5 Flash
4.3
GPT-5.5
4.0
Solar Pro 3
2.5
Claude Opus 4.8
2.4
KT Mi:dm 2.0 Base Instruct
1.0

Avg Token Usage

K-EXAONE-236B-A23B
1484.7
KT Mi:dm 2.0 Base Instruct
969.0
Gemini 3.5 Flash
892.3
Solar Pro 3
511.7
DeepSeek V4 Pro
400.0
Claude Opus 4.8
348.0
GPT-5.5
294.7
    Tokyo · Problem 2 · Gemini 3.5 Flash | EntropyMath