Difference between revisions of "2022 AMC 8 Problems/Problem 23"
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==Problem== | ==Problem== | ||
| − | A <math>\triangle</math> or <math>\bigcirc</math> is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three <math>\triangle</math>s in a line. | + | A <math>\triangle</math> or <math>\bigcirc</math> is placed in each of the nine squares in a <math>3</math>-by-<math>3</math> grid. Shown below is a sample configuration with three <math>\triangle</math>s in a line. |
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<asy> | <asy> | ||
//diagram by kante314 | //diagram by kante314 | ||
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draw(shift(2.5,2.38) * equi); | draw(shift(2.5,2.38) * equi); | ||
</asy> | </asy> | ||
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How many configurations will have three <math>\triangle</math>s in a line and three <math>\bigcirc</math>s in a line? | How many configurations will have three <math>\triangle</math>s in a line and three <math>\bigcirc</math>s in a line? | ||
<math>\textbf{(A) } 39 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 96</math> | <math>\textbf{(A) } 39 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 96</math> | ||
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==Solution== | ==Solution== | ||
Revision as of 20:09, 28 January 2022
Problem
A 
or 
is placed in each of the nine squares in a 
-by- grid. Shown below is a sample configuration with three s in a line.
![[asy] //diagram by kante314 size(3.3cm); defaultpen(linewidth(1)); real r = 0.37; path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle; draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,1)--(3,1)--(3,2)--(0,2)--cycle); draw((1,0)--(1,3)--(2,3)--(2,0)--cycle); draw(circle((3/2,5/2),1/3)); draw(circle((5/2,1/2),1/3)); draw(circle((3/2,3/2),1/3)); draw(shift(0.5,0.38) * equi); draw(shift(1.5,0.38) * equi); draw(shift(0.5,1.38) * equi); draw(shift(2.5,1.38) * equi); draw(shift(0.5,2.38) * equi); draw(shift(2.5,2.38) * equi); [/asy]](http://latex.artofproblemsolving.com/f/7/9/f79f74a5bed66611fe027a9dd5ab0948b3a033ac.png)
How many configurations will have three s in a line and three s in a line?
Solution
See Also
| 2022 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
