2023 AIME II Problem 6

Question

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.

Answer

The three squares plus the missing fourth square tile a

2 \times 2

square, so the midpoint can fail only by landing in the missing square

Failure needs one point in the top square and one in the right square, and then each coordinate condition, like

x_1 + x_2 \gt 2,

holds with probability

\frac{1}{2}

Place the region as

[0,1]^2 \cup \bigl([0,1] \times [1,2]\bigr)

\cup \bigl([1,2] \times [0,1]\bigr),

so it is the

2 \times 2

square with the top-right unit square removed. Both coordinates of the midpoint are averages of numbers in

[0, 2],

so the midpoint always lies in the

2 \times 2

square; it fails to lie in the region exactly when it lands in the missing square, i.e. when

x_A + x_B \gt 2

and

y_A + y_B \gt 2.

If neither point is in the right square, then

x_A + x_B \le 2;

if neither is in the top square, then

y_A + y_B \le 2.

So failure requires one point in the top square and the other in the right square, which happens with probability

2 \cdot \frac{1}{3} \cdot \frac{1}{3} = \frac{2}{9}.

In that case, one

x

-coordinate is uniform on

[0,1]

and the other on

[1,2],

so

x_A + x_B \gt 2

with probability

\frac{1}{2},

and independently

y_A + y_B \gt 2

with probability

\frac{1}{2}.

The failure probability is

\frac{2}{9} \cdot \frac{1}{4} = \frac{1}{18},

so the desired probability is

\frac{17}{18}

and

m + n = 17 + 18 = 35.

Problem 6 in other years