Difference between revisions of "2021 AMC 10B Problems/Problem 23"
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| − | + | ==Problem== | |
| − | + | A square with side length <math>8</math> is colored white except for <math>4</math> black isosceles right triangular regions with legs of length <math>2</math> in each corner of the square and a black diamond with side length <math>2\sqrt{2}</math> in the center of the square, as shown in the diagram. A circular coin with diameter <math>1</math> is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as <math>\frac{1}{196}(a+b\sqrt{2}+\pi)</math> | |
| − | + | [asy] | |
| − | + | /* Made by samrocksnature */ | |
| − | + | draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); | |
| − | + | fill((2,0)--(0,2)--(0,0)--cycle, black); | |
| − | . | + | fill((6,0)--(8,0)--(8,2)--cycle, black); |
| + | fill((8,6)--(8,8)--(6,8)--cycle, black); | ||
| + | fill((0,6)--(2,8)--(0,8)--cycle, black); | ||
| + | fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); | ||
| + | filldraw(circle((2.6,3.31),0.47),gray); | ||
| + | [/asy] | ||
| + | |||
| + | <math>\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72</math> | ||
| + | ==Solution== | ||
Revision as of 18:43, 11 February 2021
Problem
A square with side length 
is colored white except for 
black isosceles right triangular regions with legs of length 
in each corner of the square and a black diamond with side length 
in the center of the square, as shown in the diagram. A circular coin with diameter 
is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as 
[asy]
/* Made by samrocksnature */
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
fill((2,0)--(0,2)--(0,0)--cycle, black);
fill((6,0)--(8,0)--(8,2)--cycle, black);
fill((8,6)--(8,8)--(6,8)--cycle, black);
fill((0,6)--(2,8)--(0,8)--cycle, black);
fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black);
filldraw(circle((2.6,3.31),0.47),gray);
[/asy]
