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prove a rectangle it´s a square

Source: 2022 Centroamerican and Caribbean Mathematical Olympiad, P4

December 1, 2022

geometryrectanglesquaretangent circles

Problem Statement

Let

A_1A_2A_3A_4

be a rectangle and let

S_1,S_2,S_3,S_4

four circumferences inside of the rectangle such that

S_k

and

S_{k+1}

are tangent to each other and tangent to the side

A_kA_{k+1}

for

k=1,2,3,4

, where

A_5=A_1

and

S_5=S_1

. Prove that

A_1A_2A_3A_4

is a square.