2023 AIME II Problems/Problem 6

Problem

Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ $[asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy]$

Solution 1

We proceed by calculating the complement.

Note that the only configuration of the 2 points that makes the midpoint outside of the L shape is one point in the top square, and one in the right square. This occurs with $\frac{2}{3} \cdot \frac{1}{3}$ probability.

Let the topmost coordinate have value of: $(x_1,y_1+1)$ , and rightmost value of: $(x_2+1,y_2)$ .

The midpoint of them is thus: $\left(\frac{x_1+x_2+1}{2}, \frac{y_1+y_2+1}{2} \right)$

It is clear that $x_1, x_2, y_1, y_2$ are all between 0 and 1. For the midpoint to be outside the L-shape, both coordinates must be greater than 1, thus: $\frac{x_1+x_2+1}{2}>1$ $x_1+x_2>1$

By symmetry this has probability $\frac{1}{2}$ . Also by symmetry, the probability the y-coordinate works as well is $\frac{1}{2}$ . Thus the probability that the midpoint is outside the L-shape is: $\frac{2}{3} \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{1}{2}$ $\frac{1}{18}$

We want the probability that the point is inside the L-shape however, which is $1-\frac{1}{18}=\frac{17}{18}$ , yielding the answer of $17+18=\boxed{35}$ ~SAHANWIJETUNGA

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2023 AIME II Problems/Problem 6

Problem

Solution 1