[THOUGHT] To solve this problem, we need to find the probability ( P(A) ) that at least one of the students ( S_1, S_2, ) or ( S_3 ) can solve the problem. We are given probabilities for events ( B, C, ) and ( D ), which relate to the students' abilities to solve the problem under certain conditions.

1. Event Definitions:

   • ( A ): At least one student can solve the problem.
   • ( B ): ( S_1 ) can solve the problem given that neither ( S_2 ) nor ( S_3 ) can solve it. This is a conditional probability: ( P(B) = P(S_1 \text{ solves} \mid S_2 \text{ and } S_3 \text{ do not solve}) ).
   • ( C ): ( S_2 ) can solve the problem and ( S_3 ) cannot. This is ( P(S_2 \text{ solves and } S_3 \text{ does not solve}) ).
   • ( D ): ( S_3 ) can solve the problem. This is ( P(S_3 \text{ solves}) ).

2. Given Probabilities:

   • ( P(B) = \frac{1}{2} )
   • ( P(C) = \frac{1}{10} )
   • ( P(D) = \frac{1}{12} )

3. Approach:

   • First, express ( P(B) ) in terms of the probabilities of the students solving the problem. Let ( p_1, p_2, p_3 ) be the probabilities that ( S_1, S_2, S_3 ) can solve the problem, respectively.
   • ( P(B) = \frac{P(S_1 \text{ solves})}{P(S_2 \text{ and } S_3 \text{ do not solve})} = \frac{p_1}{(1 - p_2)(1 - p_3)} = \frac{1}{2} ).
   • ( P(C) = p_2 (1 - p_3) = \frac{1}{10} ).
   • ( P(D) = p_3 = \frac{1}{12} ).

4. Solve for ( p_2 ):

   • From ( P(C) = p_2 (1 - p_3) = \frac{1}{10} ) and ( p_3 = \frac{1}{12} ), we can solve for ( p_2 ).

5. Solve for ( p_1 ):

   • Using ( P(B) = \frac{p_1}{(1 - p_2)(1 - p_3)} = \frac{1}{2} ), we can solve for ( p_1 ).

6. Calculate ( P(A) ):

   • ( P(A) = 1 - P(\text{None solve}) = 1 - (1 - p_1)(1 - p_2)(1 - p_3) ).

7. Simplify ( P(A) ) to the form ( \frac{p}{q} ) and compute ( p^2 + q^2 ).

[/THOUGHT] [PYTHON] from fractions import Fraction

Given probabilities

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

p3 is P(D)

p3 = P_D

Solve for p2 using P(C) = p2 * (1 - p3)

p2 = P_C / (1 - p3)

Solve for p1 using P(B) = p1 / ((1 - p2) * (1 - p3))

p1 = P_B * (1 - p2) * (1 - p3)

Calculate P(A) = 1 - (1 - p1)(1 - p2)(1 - p3)

P_A = 1 - (1 - p1) * (1 - p2) * (1 - p3)

Simplify P_A to p/q

p = P_A.numerator q = P_A.denominator

Compute p^2 + q^2

result = p2 + q2

print(result) [/PYTHON]

[THOUGHT] The Python output confirms that ( p^2 + q^2 = 65685841 ). This is the final answer based on the calculations. [/THOUGHT] \boxed{65685841}