2020 AIME II Problem 2

Question

Let

P

be a point chosen uniformly at random in the interior of the unit square with vertices at

(0, 0),

(1, 0),

(1, 1),

and

(0, 1).

The probability that the slope of the line determined by

P

and the point

\left(\frac{5}{8}, \frac{3}{8}\right)

is greater than or equal to

\frac{1}{2}

can be written as

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

Answer: 171

Concepts:geometric probability slope trapezoid

Difficulty rating: 2110

Small hint:

For

P = (x, y)

with

x \gt \frac{5}{8},

the condition is

y \ge \frac{x}{2} + \frac{1}{16};

for

x \lt \frac{5}{8}

the inequality direction flips

Big hint:

The favorable region is two trapezoids meeting at

\left(\frac{5}{8}, \frac{3}{8}\right);

find each area and add

Solution:

Let

Q = \left(\frac{5}{8}, \frac{3}{8}\right)

and

P = (x, y).

The slope condition

\frac{y - \frac{3}{8}}{x - \frac{5}{8}} \ge \frac{1}{2}

becomes

y \ge \frac{x}{2} + \frac{1}{16}

when

x \gt \frac{5}{8},

and

y \le \frac{x}{2} + \frac{1}{16}

when

x \lt \frac{5}{8}

(multiplying by the negative quantity

x - \frac{5}{8}

reverses the inequality).

For

x \gt \frac{5}{8},

the region above the line

y = \frac{x}{2} + \frac{1}{16}

inside the square is a trapezoid with parallel vertical sides of lengths

\frac{5}{8}

(at

x = \frac{5}{8}

) and

\frac{7}{16}

(at

x = 1

) and width

\frac{3}{8},

with area

\frac{3}{8} \cdot \frac{\frac{5}{8} + \frac{7}{16}}{2} = \frac{51}{256}.

For

x \lt \frac{5}{8},

the region below the line is a trapezoid with parallel sides

\frac{1}{16}

(at

x = 0