Problem 25
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A point $P$ is chosen at random inside square $ABCD$. The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\frac{a + b \pi - c \sqrt{d}}{e}$, where $a, b, c, d,$ and $e$ are positive integers, $\text{gcd}(a, b, c, e) = 1$, and $d$ is not divisible by the square of a prime. What is $a+b+c+d+e$?
$\textbf{(A) }25 \qquad \textbf{(B) }26 \qquad \textbf{(C) }27 \qquad \textbf{(D) }28 \qquad \textbf{(E) } 29 \qquad$
Difficulty: 66/100 — A typical geometrical probability problem.
Core Concepts: geometric probability
Challenges: Tackling both of the cases for when either of the two side lengths are the shortest side length of $\triangle APB$ and combining into one. Working with a condition with respect to two other values.
Techniques: Splitting the problem into two cases of when $BP$ or $AB$ are the shortest side lengths, and adding the areas of each region together.
Error-prone Steps: Forgetting to subtract an extra constant when calculating the probability.
Ideal Time:
Experienced: ≤ 3 min
Intermediate: ≤ 6 min
Beginner: would not solve