Question

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 $?$

$0.3$

$0.4$

$0.5$

$0.6$

$0.7$

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Written 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.

2020 AMC 10A Problem 16

Video solution:

Written solution:

Problem 16 in Other Years