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Miklós Schweitzer 2002, Problem 4

Source: Miklós Schweitzer 2002

July 30, 2016

college contestsMiklos Schweitzeralgebra

Problem Statement

For a given natural number

n

, consider those sets

A\subseteq \mathbb{Z}_n

for which the equation

xy=uv

has no other solution in the residual classes

x,y,u,v\in A

than the trivial solutions

x=u

,

y=v

and

x=v

,

y=u

. Let

g(n)

be the maximum of the size of such sets

A

. Prove that

\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1

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