1

Part of 1990 Irish Math Olympiad

Problems(2)

Number of Rectangles in the Plane

Source: 1990 IrMO Paper 1 Problem 1

Given a natural number

n

, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range

0

n

, and whose sides are parallel to the axes.

rectanglecombinatorics

Irish 1990 paper 2 #1

Source: 2nd Irish Mathematical Olympiad

n>3

be a natural number . Prove that

\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.

inequalities proposedinequalities