2015 AMC 12A Problem 23

Question

Let

S

be a square of side length

1.

Two points are chosen independently at random on the sides of

S.

The probability that the straight-line distance between the points is at least

\dfrac12

is

\dfrac{a - b\pi}{c},

where

a,

b,

and

c

are positive integers and

\gcd(a, b, c) = 1.

What is

a + b + c?

59

60

61

62

63

Condition on whether the two points lie on the same side, opposite sides, or adjacent sides

The adjacent-sides case produces the

\pi

term via a quarter-circle of radius

\dfrac12

The second point is on the same side as the first with probability

\dfrac14,

on the opposite side with probability

\dfrac14,

and on an adjacent side with probability

\dfrac12.

Opposite sides: the distance is at least

1 \ge \dfrac12

always, probability

1.

Same side: for points

(a, 0)

and

(b, 0),

the condition

|a - b| \ge \dfrac12

has probability

\dfrac14.

Adjacent sides: for points

(a, 0)

and

(0, b),

the condition

\sqrt{a^2 + b^2} \ge \dfrac12

is the region outside a quarter-circle of radius

\dfrac12,

with probability

1 - \dfrac14\pi\left(\dfrac12\right)^2 = 1 - \dfrac{\pi}{16}.

The total probability is

\begin{aligned} &\dfrac14\cdot 1 + \dfrac14\cdot\dfrac14 \\ &\quad {}+ \dfrac12\left(1 - \dfrac{\pi}{16}\right) \\ &\quad = \dfrac{26 - \pi}{32}. \end{aligned}

Thus

a + b + c = 26 + 1 + 32 = 59.

Thus, the correct answer is A.

Problem 23 in other years