Problem Analysis #27

Gemini 3.5 Flash · Run 1

Correct

8.15s

1,499 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.

A maximum area has value $100\sqrt8$ . Write this value in the form

\frac{p\sqrt q}{r},

where $p,q,r$ are positive integers, $q$ is squarefree, and $\gcd(p,r)=1$ . Find $p+q+r$ .

Ground Truth

203

Recorded model response / solution

Reasoning Process

To write the value $100$$\sqrt{8}$$$ in the form$ \frac{p\sqrt{q}}{r}$, we first simplify the expression:

100\sqrt{8} = 100\sqrt{4 \times 2} = 100 \times 2\sqrt{2} = 200\sqrt{2}

We can write this as:

\frac{200\sqrt{2}}{1}

Comparing this to the form $\frac{p\sqrt{q}}{r}$ , we have:

$p = 200$
$q = 2$
$r = 1$

We check the given conditions:

$p, q, r$ are positive integers ( $200, 2, 1$ are positive integers).
$q = 2$ is squarefree.
$\gcd(p, r) = \gcd(200, 1) = 1$ .

All conditions are satisfied. Now, we find the sum $p+q+r$ :

p+q+r = 200 + 2 + 1 = 203

\boxed{203}

Final Model Output

203

Comparative Analysis (P27) (Average across runs)

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

Avg Time Taken

K-EXAONE-236B-A23B

31.6

DeepSeek V4 Pro

10.1

Gemini 3.5 Flash

7.1

GPT-5.5

3.6

Solar Pro 3

2.6

Claude Opus 4.8

1.8

KT Mi:dm 2.0 Base Instruct

0.6

Avg Token Usage

K-EXAONE-236B-A23B

1933.7

Gemini 3.5 Flash

1482.0

KT Mi:dm 2.0 Base Instruct

1001.0

DeepSeek V4 Pro

724.3

Solar Pro 3

584.7

Claude Opus 4.8

312.0

GPT-5.5

309.3