2009 AMC 12B Problem 22

Question

Parallelogram

ABCD

has area

1{,}000{,}000.

Vertex

A

is at

(0, 0)

and all other vertices are in the first quadrant. Vertices

B

and

D

are lattice points on the lines

y = x

and

y = kx

for some integer

k \gt 1,

respectively. How many such parallelograms are there?

49

720

784

2009

2048

With

B = (b, b)

and

D = (d, kd),

the area of the parallelogram is

(k - 1)bd

You need

(k - 1)bd = 10^6 = 2^6 \cdot 5^6

; count ordered factorizations into three positive integers

Let

B = (b, b)

and

D = (d, kd)

with

b, d, k

positive integers and

k \gt 1.

The area is

(k - 1)bd = 1{,}000{,}000 = 2^6 \cdot 5^6.

Each parallelogram corresponds to an ordered triple

(k - 1, b, d)

of positive integers with product

2^6 \cdot 5^6.

The six

2

’s distribute among the three factors in

\binom{6 + 2}{2} = 28

ways, and likewise the six

5

’s in

28

ways, giving

28^2 = 784.

Thus, the correct answer is C.

Problem 22 in other years