2016 AMC 10B Problem 14

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:lattice point counting shapes in figures casework

Difficulty rating: 1970

Small hint:

For each square size, count possible top-left lattice points

Big hint:

Only side lengths

1,2,3

can fit

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

Answer: D

Difficulty rating: 1970

Small hint:

Big hint:

Solution:

Problem 14 in other years