Problem Analysis #10

DeepSeek V4 Pro · Run 1

Incorrect

45.90s

2,998 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.

Let $a>0$ , and consider the parabola $y=x^2$ . Starting from $P(a,a^2)$ , let $Q\ne P$ be the other intersection of the normal line to the parabola at $P$ with the parabola. Let $R\ne Q$ be the other intersection of the normal line at $Q$ with the parabola.

If the minimum possible $x$ -coordinate of $R$ is $M$ , write $M^2=p/q$ in lowest terms and find $p+q$ .

Ground Truth

Recorded model response / solution

Reasoning Process

The minimum possible $x$ -coordinate of $R$ is $M = \frac{3\sqrt{2}}{4}$ . Then $M^2 = \frac{9}{8}$ , giving $p=9$ , $q=8$ , and $p+q=17$ .

\boxed{17}

Final Model Output

Comparative Analysis (P10) (Average across runs)

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

Avg Time Taken

K-EXAONE-236B-A23B

77.5

DeepSeek V4 Pro

55.9

Solar Pro 3

41.5

Gemini 3.5 Flash

13.5

GPT-5.5

11.2

Claude Opus 4.8

9.0

KT Mi:dm 2.0 Base Instruct

3.3

Avg Token Usage

K-EXAONE-236B-A23B

11966.0

Solar Pro 3

6444.7

DeepSeek V4 Pro

3794.7

Gemini 3.5 Flash

3138.3

KT Mi:dm 2.0 Base Instruct

1946.7

Claude Opus 4.8

1157.7

GPT-5.5

811.0