Question

Consider the parallelogram with vertices $(10, 45),$ $(10, 114),$ $(28, 153),$ and $(28, 84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer

Solution:

A parallelogram is symmetric under the $180^\circ$ rotation about its center, so any line through the center cuts it into two pieces that are swapped by that rotation, hence congruent. The center is the midpoint of a diagonal: $\left(\frac{10 + 28}{2}, \frac{45 + 153}{2}\right) = (19, 99).$

The line through the origin and $(19, 99)$ has slope $\frac{99}{19},$ and $\gcd(99, 19) = 1,$ so $m + n = 99 + 19 = 118.$

1999 AIME Problem 2

Solution:

Problem 2 in Other Years