Difference between revisions of "2002 AMC 8 Problems/Problem 23"
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==Video Solution== | ==Video Solution== | ||
https://youtu.be/WF3mJoFCXCM Soo, DRMS, NM | https://youtu.be/WF3mJoFCXCM Soo, DRMS, NM | ||
| + | |||
| + | https://www.youtube.com/watch?v=rJnc0Tes3LU ~David | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2002|num-b=22|num-a=24}} | {{AMC8 box|year=2002|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 20:03, 15 April 2023
Contents
Problem
A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?
![[asy] /* AMC8 2002 #23 Problem */ fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey); fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey); fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey); fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey); draw((0,0)--(0,11)--(11,11)); for ( int x = 1; x < 11; ++x ) { draw((x,11)--(x,0), linetype("4 4")); } for ( int y = 1; y < 11; ++y ) { draw((0,y)--(11,y), linetype("4 4")); } clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy]](http://latex.artofproblemsolving.com/9/1/0/91026905d520b2d9e85b96a8902123543951d953.png)
Solution
The same pattern is repeated for every 
tile. Looking closer, there is also symmetry of the top 
square, so the fraction of the entire floor in dark tiles is the same as the fraction in the square. Counting the tiles, there are 
dark tiles, and 
total tiles, giving a fraction of 
.
Video Solution
https://youtu.be/WF3mJoFCXCM Soo, DRMS, NM
https://www.youtube.com/watch?v=rJnc0Tes3LU ~David
See Also
| 2002 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. 
