Difference between revisions of "2021 AMC 10B Problems/Problem 23"
Pi is 3.14 (talk | contribs) (→Video Solution by OmegaLearn (Similar Triangles and Area Calculations)) |
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==Problem== | ==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> | + | 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>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? |
<asy> | <asy> | ||
/* Made by samrocksnature */ | /* Made by samrocksnature */ | ||
Revision as of 03:39, 12 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 
, where 
and 
are positive integers. What is 
?
![[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]](http://latex.artofproblemsolving.com/7/8/2/782e019eee5da15727a6caf03020a1dfd545b11d.png)
Video Solution by OmegaLearn (Similar Triangles and Area Calculations)
~ pi_is_3.14
| 2021 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
