HMMT February, 2015

Geometry Difficulty 5.3 AIME, harder Prove it United States

Problem:

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$ , $(2,0)$ , $(2,1)$ , and $(0,1)$ . $R$ can be divided into two unit squares, as shown.

Pro selects a point $P$ uniformly at random in the interior of $R$ . Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

Solution

Solution:

Answer: $\frac{3}{4}$

Precisely the middle two (of four) regions satisfy the problem conditions, and it's easy to compute the (fraction of) areas as $\frac{3}{4}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.