[THOUGHT] To solve the problem, we need to determine the probability $P(A)$ that at least one of the three students can solve the problem, given the probabilities of specific events. We start by defining the joint probabilities for all possible scenarios where each student can or cannot solve the problem. Using the given probabilities $P(B) = \frac{1}{2}$, $P(C) = \frac{1}{10}$, and $P(D) = \frac{1}{12}$, we derive relationships between these probabilities and solve for the unknowns.

Key steps include:

1. Recognizing that $P(B)$ implies a conditional probability leading to $P_8 = P_4$.
2. Using the total probability sum to express other probabilities in terms of $P_4$.
3. Solving for $P_4$ and subsequently $P_8$, which allows us to compute $P(A) = 1 - P_8$.

After verifying all constraints and ensuring the probabilities sum to 1, we find $P(A) = \frac{71}{120}$. The values $p = 71$ and $q = 120$ are coprime, leading to $p^2 + q^2 = 71^2 + 120^2$.

[/THOUGHT] [PYTHON]

Calculate p^2 + q^2 where p = 71 and q = 120

p = 71 q = 120 result = p2 + q2 print(result) [/PYTHON]

[THOUGHT] The Python code output confirms that $p^2 + q^2 = 19441$, where $p = 71$ and $q = 120$ are coprime integers representing $P(A) = \frac{71}{120}$. This result is derived from the given probabilities and verified through the code execution. [/THOUGHT] \boxed{19441}