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IMO LongList 1967, Hungary 4

Source: IMO LongList 1967, Hungary 4

December 16, 2004

analytic geometrygeometrygeometric inequalitycirclesIMO ShortlistIMO Longlist

Problem Statement

Let

k_1

and

k_2

be two circles with centers

O_1

and

O_2

and equal radius

r

such that

O_1O_2 = r

. Let

A

and

B

be two points lying on the circle

k_1

and being symmetric to each other with respect to the line

O_1O_2

. Let

P

be an arbitrary point on

k_2

. Prove that

PA^2 + PB^2 \geq 2r^2.