2020 AMC 10A Problem 16

A point is chosen at random within the square in the coordinate plane whose vertices are

(0, 0),

(2020, 0),

(2020, 2020),

and

(0, 2020).

The probability that the point is within

d

units of a lattice point is

\tfrac{1}{2}.

(A point

(x, y)

is a lattice point if

x

and

y

are both integers.) What is

d

to the nearest tenth?

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Solution

For

d<\dfrac12

, the points within

d

of lattice points occupy, in each unit square, four quarter-circles whose total area is

\pi d^2

. The enormous square is tiled by unit squares, so the desired probability is

\pi d^2

. Setting

\pi d^2=\dfrac12

gives

d=\sqrt{\dfrac{1}{2\pi}}\approx0.399

, which rounds to

0.4

. Thus, B is the correct answer.