MathDB

n-element set

Source: 2012 China TST Test 2 p2

March 19, 2012

pigeonhole principlefloor functionceiling functioninequalitiesfunctionalgebradifference of squares

Problem Statement

Prove that there exists a positive real number

C

with the following property: for any integer

n\ge 2

and any subset

X

of the set

\{1,2,\ldots,n\}

such that

|X|\ge 2

, there exist

x,y,z,w \in X

(not necessarily distinct) such that

0<|xy-zw|<C\alpha ^{-4}

where

\alpha =\frac{|X|}{n}