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 $x=5.1?$

$30$

$41$

$45$

$50$

$57$

Answer: D

Concepts:counting shapes in figures lattice point casework

Difficulty rating: 1630

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.

2016 AMC 12B Problem 11

Answer: D

Difficulty rating: 1630

Solution:

Problem 11 in Other Years