Problem Analysis #27

KT Mi:dm 2.0 Base Instruct · Run 1

Correct

0.61s

1,001 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 solve the problem, we start by expressing the given value $100\sqrt{8}$ in the required form $\frac{p\sqrt{q}}{r}$ .

Simplify $\sqrt{8}$ :

\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}.

Substitute back into the original expression:

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

Rewrite $200\sqrt{2}$ in the form $\frac{p\sqrt{q}}{r}$ :
- Here, $p = 200$ , $q = 2$ , and $r = 1$ (since there is no denominator in the simplified form).
- Check the conditions:
  - $p = 200$ , $q = 2$ , and $r = 1$ are all positive integers.
  - $q = 2$ is squarefree.
  - $\gcd(200, 1) = 1$ .
Calculate $p + q + r$ :

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

Final Answer: $\boxed{203}$

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