1997 AIME Problem 2

Question

The nine horizontal and nine vertical lines on an

8 \times 8

checkerboard form

r

rectangles, of which

s

are squares. The number

\frac{s}{r}

can be written in the form

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

A rectangle is determined by choosing

2

of the

9

horizontal lines and

2

of the

9

vertical lines

There are

(9 - k)^2

squares of size

k \times k;

sum over

k = 1

to

8

A rectangle is determined by choosing two of the nine horizontal lines and two of the nine vertical lines, so

r = \binom{9}{2}^2 = 36^2 = 1296.

A

k \times k

square can be placed in

(9 - k)^2

positions, so

\begin{aligned} s &= \sum_{k=1}^{8} (9 - k)^2 \\ &= 8^2 + 7^2 + \cdots + 1^2 \\ &= \frac{8 \cdot 9 \cdot 17}{6} = 204. \end{aligned}

Then

\frac{s}{r} = \frac{204}{1296} = \frac{17}{108},

which is in lowest terms, so

m + n = 17 + 108 = 125.

Problem 2 in other years