2016 AMC 12B Problem 11

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

Small hint:

The region sits below

y=\pi x;

note

3\lt\pi\lt4,

6\lt2\pi\lt7,

and

9\lt3\pi\lt10

Big hint:

In each vertical strip

k\le x\le k+1,

count the

1\times1,

2\times2,

and

3\times3

squares that fit below the line, then add them up

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

Small hint:

Big hint:

Solution:

Problem 11 in other years