2015 AMC 10A Problem 25

A uniformly random point on the boundary amounts to picking one of the four sides (each with probability $\frac14$) and then a uniform position along it. For two points, the sides are the same with probability $\frac14$, adjacent with probability $\frac12$, and opposite with probability $\frac14$.

Same side. Positions $u, v \in [0,1]$; we need $|u - v| \ge \frac12$. In the unit square of $(u, v)$ this region is two right triangles with legs $\frac12$, total area $2 \cdot \frac18 = \frac14$.

Adjacent sides. Put the shared corner at the origin, so the points are $(u, 0)$ and $(0, v)$ and the distance is $\sqrt{u^2 + v^2}$. The failing region $u^2 + v^2 < \frac14$ is a quarter disc of radius $\frac12$, area $\frac14 \cdot \pi \cdot \frac14 = \frac{\pi}{16}$. Success probability $1 - \frac{\pi}{16}$.

Opposite sides. The distance is at least the side length $1 > \frac12$, so success probability $1$.

Combine with the weights:
$$\frac14 \cdot \frac14 + \frac12\left(1 - \frac{\pi}{16}\right) + \frac14 \cdot 1 = \frac{1}{16} + \frac12 + \frac14 - \frac{\pi}{32} = \frac{13}{16} - \frac{\pi}{32} = \frac{26 - \pi}{32}.$$
Here $a = 26$, $b = 1$, $c = 32$ with $\gcd(26, 1, 32) = 1$, so $a + b + c = 59$.

The answer is $\boxed{\textbf{(A)}\ 59}$.

When the sample space splits into structurally different pieces (which pair of sides), condition on the piece and weight each conditional probability by the chance of that piece. Inside each piece the two positions form a unit square of parameters and the probability is an area ratio. The $\pi$ in the answer form is a strong hint that one case involves a circle: points within distance $\frac12$ of a corner fill a quarter disc.

Compute the complement, the probability the distance is less than $\frac12$: same side contributes $\frac14 \cdot \frac34 = \frac{3}{16}$, adjacent sides contribute $\frac12 \cdot \frac{\pi}{16} = \frac{\pi}{32}$, opposite sides contribute $0$. Total $\frac{6 + \pi}{32}$, so the desired probability is $1 - \frac{6 + \pi}{32} = \frac{26 - \pi}{32}$.

• Averaging the three side-pair cases equally instead of weighting them $\frac14, \frac12, \frac14$, or getting $\frac12$ instead of $\frac14$ for the same-side case.
• Using a half disc or full disc ($\frac{\pi}{8}$ or $\frac{\pi}{4}$) for the adjacent-sides failure region instead of the quarter disc inside the corner.
• Reporting $13 + 1 + 16$ or another unreduced or mis-shaped form; the answer must be written as a single fraction $\frac{a - b\pi}{c}$ with $b$ an integer.