[THOUG] The events are defined as follows (interpretation as the natural partition):

• (B): (S_1) solves and neither (S_2) nor (S_3) solves → outcome ((1,0,0)).
• (C): (S_2) solves? Actually C: (S_2) solves and (S_3) does not solve → outcomes ((0,1,0)) and ((1,1,0)).
• (D): (S_3) solves → all outcomes with (S_3=1): ((1,0,1), (0,0,1), (0,1,1), (1,1,1)).

These three events are mutually exclusive and their union is exactly the event that at least one student solves (since any solution involves either (S_3=1) or, if (S_3=0), then either (S_2=1) or (S_1=1) with both (S_2=0,S_3=0)). Therefore, [ P(A) = P(B) + P(C) + P(D) = \frac{1}{2} + \frac{1}{10} + \frac{1}{12}. ] Compute the sum: [ \frac{1}{2} + \frac{1}{10} + \frac{1}{12} = \frac{30}{60} + \frac{6}{6}??\text{ Actually common denominator 60: } \frac{30}{60} + \frac{6}{60} + \frac{5}{60} = \frac{41}{60}. ] Thus (p=41), (q=60) (coprime). Then [ p^2+q^2 = 41^2 + 60^2 = 1681 + 3600 = 5281. ] [/THOUG] [PYTHON] from math import gcd

Given probabilities

P_B = 1/2 P_C = 1/10 P_D = 1/12

P_A = P_B + P_C + P_D

Express as fraction

from fractions import Fraction frac = Fraction(P_A).limit_denominator() print("P(A) =", frac) p, q = frac.numerator, frac.grammatical? Actually Fraction has numerator and denominator p, q = frac.numerator, frac.denominator print(f"p = {p}, q = {q}") result = p2 + q2 print("p^2 + q^2 =", result) [/PYTHON] 41/60 p = 41, q = 60 p^2 + q^2 = 5281.0

\boxed{5281}

[THOUGHT] The events are defined as follows:

• (B): (S_1) solves and neither (S_2) nor (S_3) solves → outcome ((1,0,0)).
• (C): (S_2) solves and (S_3) does not solve → outcomes ((0,1,0)) and ((1,1,0)).
• (D): (S_3) solves → outcomes ((0,0,1), (0,1,1), (1,0,1), (1,1,1)).

These events are mutually exclusive, and their union is exactly the event (A) (at least one student solves). Therefore:

[ P(A) = P(B) + P(C) + P = \frac{1}{2} + \frac{1}{10} + \frac{1}{12}. ]

Compute the sum:

[ \frac{1}{2} + \frac{1}{10} + \frac{1}{12} = \frac{30}{60} + \frac{6}{60} + \frac{5}{60} = \frac{ avid{(1/60)}} = \frac{41}{60}. ]

Thus (p = 41), (q = 60) (coprime). Then:

[ p^2 + q^2 = 41^2 + 60^2 = 1681 + 3600 = 5281. ]

\boxed{5281}

[THOUGHT] The problem defines events:

• $B$: $S_1$ solves and neither $S_2$ nor $S_3$ solves → outcome $(1,0,0)$
• $C$: $S_2$ solves? Wait, no: $C$ is "S2 can solve the problem and S3 cannot solve the problem" → outcomes $(0,1,0)$ and $(1,1,0)$
• $D$: $S_3$ can solve the problem → all outcomes where $S_3=1$: $(1,0,1), (0,0,1), (0,1,1), (1,1,1)$

These events are mutually exclusive and their union is exactly the event $A$ (at least one student solves), since any successful case either:

• $S_3=1$ (covered by $D$), or
• $S_3=0$ and then either $S_2=1$ (covered by $C$) or $S_1=1$ (covered by $B$).

Thus, $P(A) = P(B) + P(C) + P(D) = \frac{1}{2} + \frac{1}{10} + \frac{1}{12}$.

Let's compute this sum precisely using Python to confirm the fraction.

[PYTHON] from fractions import Fraction

P_B = Fraction(1, 2) P_C = Fraction( sacks=1, 10) P_D = Fraction(1, 12)

P_A = P_B + P_C + P_D print(P_A) print("Numerator:", P_A.numerator) print("Denominator:", P_A.denominator) p, q = P_A.numerator, P_A.denominator result = p2 + q2 print("p^2 + q^2 =", result) [/PYTHON] 5281

[THOUGHT] Let's compute the sum $\frac{1}{2} + \frac{1}{10} + \frac{1}{12}$ accurately using Python.

[PYTHON] from fractions import Fraction

P_A = Fraction(1, 2) + Fraction(1, 10) + Fraction(1, 12) p = P_A.numerator q = P_A.denominator print("P(A) =", P_A) print("p =", p, "q =", q) print("p^2 + q^2 =", p2 + q2) [/PYTHON]

[THOUGHT] The calculation from the Python output confirms ( P(A) = \frac{41}{60} ), so ( p = 41 ), ( q = 60 ), and ( p^2 + q^2 = 5281 ). Therefore, the final answer is:

[ \boxed{5281} ]