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2019 IGO Advanced P4

Source: 6th Iranian Geometry Olympiad (Advanced) P4

September 20, 2019

IGOIrangeometry

Problem Statement

Given an acute non-isosceles triangle

ABC

with circumcircle

\Gamma

.

M

is the midpoint of segment

BC

and

N

is the midpoint of arc

BC

of

\Gamma

(the one that doesn't contain

A

).

X

and

Y

are points on

\Gamma

such that

BX\parallel CY\parallel AM

. Assume there exists point

Z

on segment

BC

such that circumcircle of triangle

XYZ

is tangent to

BC

. Let

\omega

be the circumcircle of triangle

ZMN

. Line

AM

meets

\omega

for the second time at

P

. Let

K

be a point on

\omega

such that

KN\parallel AM

,

\omega_b

be a circle that passes through

B

,

X

and tangents to

BC

and

\omega_c

be a circle that passes through

C

,

Y

and tangents to

BC

. Prove that circle with center

K

and radius

KP

is tangent to 3 circles

\omega_b

,

\omega_c

and

\Gamma

.
Proposed by Tran Quan - Vietnam

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