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Many circles and a perpendicular!

Source: Iran TST 2012-Second exam-2nd day-P5

May 13, 2012

geometrycircumcircletrapezoidIran

Problem Statement

Points

A

and

B

are on a circle

\omega

with center

O

such that

\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}

. Let

C

be the circumcenter of the triangle

AOB

. Let

l

be a line passing through

C

such that the angle between

l

and the segment

OC

is

\tfrac{\pi}{3}

.

l

cuts tangents in

A

and

B

to

\omega

in

M

and

N

respectively. Suppose circumcircles of triangles

CAM

and

CBN

, cut

\omega

again in

Q

and

R

respectively and theirselves in

P

(other than

C

). Prove that

OP\perp QR

.
Proposed by Mehdi E'tesami Fard, Ali Khezeli

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