Difference between revisions of "2004 AMC 10A Problems/Problem 16"
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+ | == Video Solution == | ||
+ | https://youtu.be/0W3VmFp55cM?t=4697 | ||
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+ | ~ pi_is_3.14 | ||
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==Video Solution== | ==Video Solution== | ||
https://youtu.be/aMmF6jz6xA4 | https://youtu.be/aMmF6jz6xA4 | ||
Education, the Study of Everything | Education, the Study of Everything | ||
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==Problem== | ==Problem== |
Revision as of 23:31, 16 January 2021
Contents
Video Solution
https://youtu.be/0W3VmFp55cM?t=4697
~ pi_is_3.14
Video Solution
Education, the Study of Everything
Problem
The grid shown contains a collection of squares with sizes from
to
. How many of these squares contain the black center square?
Solution
Solution 1
Since there are five types of squares: and
We must find how many of each square contain the black shaded square in the center.
If we list them, we get that
- There is
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There are
of all
squares, containing the black square
- There is
of all
squares, containing the black square
Thus, the answer is .
Solution 2
We use complementary counting. There are only and
squares that do not contain the black square. Counting, there are
-
squares, and
squares that do not contain the black square. That gives
squares that don't contain it. There are a total of
squares possible
-
squares
-
squares
-
squares
-
squares and
-
square), therefore there are
squares that contain the black square, which is
.
See also
2004 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.