Problem Analysis #11

18.12s

3,536 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

19441

Execution Trace

Reasoning Process

[THOUGHT] I'll use Python to compute the fraction for P(A) and then p^2 + q^2. [/THOUGHT] [PYTHON] from fractions import Fraction

Given values

PB = Fraction(1, 2) PC = Fraction(1, 10) PD = Fraction(1, 12)

Let t = x4 = x8 as deduced

Total probability sum: (x1+x3+x5+x7) + (x2+x6) + (x4+x8) = 1

That is PD + PC + 2t = 1

t = Fraction(1) - (PC + PD) t /= 2

x8 = t PA = 1 - x8 # P(A) = 1 - P(none)

p, q = PA.numerator, PA.denominator result = p2 + q2 print("P(A) =", PA) print("p =", p, "q =", q) print("p^2 + q^2 =", result) [/PYTHON]

Tool Output

[PYTHON OUTPUT] P(A) = 71/120 p = 71 q = 120 p^2 + q^2 = 19441 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Using the verified Python result, (P(A) = \frac{71}{120}), so (p = 71), (q = 120), and (p^2 + q^2 = 19441). [ \boxed{19441} ]

Final Model Output

19441

Comparative Analysis (P11) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview

228.5

Llama-VARCO-8B-Instruct

51.6

Deepseek-V3.2

43.4

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

Claude-Opus-4.5

16.1

EXAONE-4.0.1-32B (high)

5.8

Avg Token Usage

Gemini-3-Pro-Preview

10703.0

Grok-4.1-fast

8474.0

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

6509.0

A.X-4.0 (72B)

5008.0

Deepseek-V3.2

4986.0

HCX-007(high)

4852.0

EXAONE-4.0.1-32B (high)

4557.0

Claude-Opus-4.5

4549.0

GPT-5.1 (high)

3536.0

Llama-VARCO-8B-Instruct

2394.0