MathDB

A subset

S

of size

n

of a plane consisting of points with both coordinates integer is given, where

n

is an odd number. The injective function

f\colon S\rightarrow S

satisfies the following: for each pair of points

A, B\in S

, the distance between points

f(A)

and

f(B)

is not smaller than the distance between points

A

and

B

. Prove there exists a point

X

such that

f(X)=X

2