2016 AMC 10B Problems/Problem 14
Contents
[hide]Problem
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line
and the line
Solution 1
The region is a right triangle which contains the following lattice points:
Squares :
Suppose that the top-right corner is
, with
. Then to include all other corners, we need
.
This produces
squares.
Squares :
Here
. To include all other corners, we need
.
This produces
squares.
Squares :
Similarly this produces
squares.
No other squares will fit in the region. Therefore the answer is .
Solution 2
The vertical line is just to the right of , the horizontal line is just under
, and the sloped line will always be above the
value of
.
This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of
,
, and
squares and getting
,
, and
respectively, and we end up with
.
Solution by Wwang
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |
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