2020 AMC 10A Problems/Problem 16

Problem

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 $?$

$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

Solution

We consider an individual one by one block.

If we draw a quarter of a circle from each corner, the area covered by the circles should be $0.5$ . Because of this, and the fact that there are four circles, we write

$4 * \frac{1}{4} * \pi r^2 = \frac{1}{2}$

Solving for $r$ , we obtain $r = \frac{1}{\sqrt{2\pi}}$ , where with $\pi \approx 3$ , we get $r = \frac{1}{\sqrt{6}}$ , and from here, we simplify and see that $r \approx 0.4 \implies \boxed{\textbf{(B) } 0.4.}$ ~Crypthes

$\textbf{Note:}$ To be more rigorous, note that $d<0.5$ since if $d\geq0.5$ then clearly the probability is greater than $\frac{1}{2}$ . This would make sure the above solution works, as if $d\geq0.5$ there is overlap with the quartercircles. $\textbf{- Emathmaster}$

Video Solution

https://youtu.be/RKlG6oZq9so

~IceMatrix

2020 AMC 10A (Problems • Answer Key • Resources)
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2020 AMC 10A Problems/Problem 16

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Problem

Solution

Video Solution

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