Problem Analysis #11

GPT-5.2 (high) · Run 1

16.90s

3,857 toks

Problem Statement

Three students $S_1$ , $S_2$ , and $S_3$ are given a problem to solve. Consider the following events: $A$ : At least one of $S_1$ , $S_2$ , and $S_3$ can solve the problem, $B$ : $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem, $C$ : $S_2$ can solve the problem and $S_3$ cannot solve the problem, $D$ : $S_3$ can solve the problem.For any event $E$ , let $P(E)$ denote the probability of $E$ . If

P(B) = \frac{1}{2}, \quad P(C) = \frac{1}{10}, \quad \text{and} \quad P(D) = \frac{1}{12},

and $P(A) = \dfrac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the value of $p^2 + q^2$ .\

Ground Truth

1465

Execution Trace

Reasoning Process

[THOUGHT] Let $X_i$ be the event that student $S_i$ can solve the problem.

Given:

$P(X_1 \mid X_2^c \cap X_3^c)=\tfrac12$ . Let $t=P(X_2^c \cap X_3^c)$ . Then $P(X_1^c \cap X_2^c \cap X_3^c)=\tfrac{t}{2}$ .
$P(X_2 \cap X_3^c)=\tfrac{1}{10}$ .
$P(X_3)=\tfrac{1}{12}$ .

Now $X_2^c \cap X_3^c$ , $X_2 \cap X_3^c$ , and $X_3$ form a partition of the sample space, so

t+\frac{1}{10}+\frac{1}{12}=1 \;\Rightarrow\; t=1-\frac{11}{60}=\frac{49}{60}.

Thus

P(\text{none solve})=P(X_1^c \cap X_2^c \cap X_3^c)=\frac{t}{2}=\frac{49}{120},

P(A)=1-\frac{49}{120}=\frac{71}{120}.

Hence $p=71,q=120$ , and $p^2+q^2=71^2+120^2$ .

[/THOUGHT] [PYTHON] from fractions import Fraction t = Fraction(1,1) - Fraction(1,10) - Fraction(1,12) P_none = t/2 P_A = Fraction(1,1) - P_none p, q = P_A.numerator, P_A.denominator print("t =", t) print("P_none =", P_none) print("P(A) =", P_A, "p,q=", p,q) print("p^2+q^2 =", pp + qq) [/PYTHON]

Tool Output

[PYTHON OUTPUT] t = 49/60 P_none = 49/120 P(A) = 71/120 p,q= 71 120 p^2+q^2 = 19441 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] From the computed $P(A)=\frac{71}{120}$ , we have $p=71$ , $q=120$ , so $p^2+q^2=19441$ .

\boxed{19441}

Final Model Output

19441

Comparative Analysis (P11) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

239.9

Gemini-3-Pro-Preview

228.5

K-EXAONE-236B-A23B

110.7

Llama-VARCO-8B-Instruct

51.6

Kanana-2-30B-Thinking

51.1

Kanana-2-30B-Thinking-2601

45.9

Deepseek-V3.2

43.4

Solar-Open-100B

40.1

HCX-007(high)

39.0

Solar-Pro-2 (31B)(high)

33.7

A.X-4.0 (72B)

26.7

GPT-5.1 (high)

18.1

Grok-4.1-fast

18.1

GPT-5.2 (high)

16.9

Claude-Opus-4.5

16.1

EXAONE-4.0.1-32B (high)

5.8

Avg Token Usage

K-EXAONE-236B-A23B

27,427

Gemini-3-Pro-Preview

10,703

Kanana-2-30B-Thinking-2601

10,671

Kanana-2-30B-Thinking

10,625

Solar-Open-100B

8,808

K-EXAONE-236B-A23B

8,679

Grok-4.1-fast

8,474

Solar-Pro-2 (31B)(high)

6,509

A.X-4.0 (72B)

5,008

Deepseek-V3.2

4,986

HCX-007(high)

4,852

EXAONE-4.0.1-32B (high)

4,557

Claude-Opus-4.5

4,549

GPT-5.2 (high)

3,857

GPT-5.1 (high)

3,536

Llama-VARCO-8B-Instruct

2,394