2016 AMC 12B Problem 11

Problem 11 of 25IntermediateGeometry Counting & Probability

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

y=\pi x,

the line

y=-0.1,

and the line

x=5.1?

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Solution

A square whose left edge is

x=k

must fit below the lowest point of

y=\pi x

over its width, namely

y=\pi k.

For side length

1,

the possible left edges

k=1,2,3,4

contribute

3+6+9+12=30

squares. For side length

2,

the left edges

k=1,2,3

contribute

2+5+8=15,

and for side length

3,

the left edges

k=1,2

contribute

1+4=5.

A square of side at least

4

cannot fit, so the total is

30+15+5=50.

Thus, the correct answer is D.

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