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A funny twist of geometry with number theory

Source: Baltic Way 2018, Problem 11

November 6, 2018

geometrynumber theory

Problem Statement

The points

A,B,C,D

lie, in this order, on a circle

\omega

, where

AD

is a diameter of

\omega

. Furthermore,

AB=BC=a

and

CD=c

for some relatively prime integers

a

and

c

. Show that if the diameter

d

of

\omega

is also an integer, then either

d

or

2d

is a perfect square.

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