Question

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 $x=5.1?$

$\ 30$

$\ 41$

$\ 45$

$\ 50$

$\ 57$

Solution:

A square must lie above $y=-0.1$ , to the left of $x=5.1$ , and below $y=\pi x$ . Since $\pi$ is a little more than $3$ , the lattice heights available at $x=1,2,3,4$ are $3,6,9,12$ , and side lengths larger than $3$ cannot fit.

Count by side length using the top-left lattice point. For side length $1$ , there are $3+6+9+12=30$ choices. For side length $2$ , there are $2+5+8=15$ choices. For side length $3$ , there are $1+4=5$ choices.

The total is $30+15+5=50$ .

Thus, the correct answer is D.

2016 AMC 10B Problem 14

Solution:

Problem 14 in Other Years