MathDB

sum of areas of rectangles equals n(n + 2)(7n + 4)/24

Source: 2006 Austria Beginners' Competition p3

October 12, 2022

combinatoricscombinatorial geometry

Problem Statement

Let

n

be an even positive integer. We consider rectangles with integer side lengths

k

and

k +1

, where

k

is greater than

\frac{n}{2}

and at most equal to

n

. Show that for all even positive integers

n

the sum of the areas of these rectangles equals

\frac{n(n + 2)(7n + 4)}{24}.