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Moldova egmo 2k17 tst

Source: MDA TST for EGMO 2017, problem 4

April 26, 2017

Problem Statement

The points

P

and

Q

are placed in the interior of the triangle

\Delta ABC

such that

m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)

and similarly for the other

2

vertices(

P

and

Q

are isogonal conjugates). Let

P_{A}

and

Q_{A}

be the intersection points of

AP

and

AQ

with the circumcircle of

CPB

, respectively

CQB

. Similarly the pairs of points

(P_{B},Q_{B})

and

(P_{C},Q_{C})

are defined. Let

PQ_{A}\cap QP_{A}=\{M_{A}\}

,

PQ_{B}\cap QP_{B}=\{M_{B}\}

,

PQ_{C}\cap QP_{C}=\{M_{C}\}

. Prove the following statements:

1.

Lines

AM_{A}

,

BM_{B}

,

CM_{C}

concur.

2.

M_{A}\in BC

,

M_{B}\in CA

,

M_{C}\in AB

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