PUMaC 2017 · 几何（A 组） · 第 2 题

PUMaC 2017 — Geometry (Division A) — Problem 2

The area of parallelogram ABCD is 51 55 and ∠ DAC is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? ′ ′ ′ ′ ′ ′ ′ ′

解析

The area of a parallelogram is the base times the height, so let AD = x , AC = . x 2 51 · 55 2 2 Because DAC is a right angle, using Pythagorean theorem, letting CD = a , a = x + . 2 x 2 4 2 2 2 2 2 4 2 Clearly, x 6 = 0, so multiplying both sides by x , x + 51 · 55 = x a , so x a − x = 51 · 55 → 2 2 2 2 2 2 x ( a − x ) = 51 · 55. For x to be an integer, x must divide 51 , so x = 1 , 3 , 17 , or 51. Testing these yields x = 17 and a = 28, so the perimeter of the parallelogram is 2(17 + 28) = 90 . Problem written by Nathan Bergman