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VMO 2022 problem 3 day 1

Source: Vietnam Mathematical Olympiad 2022 problem 3 day 1

March 4, 2022

vmogeometrycircumcircle

Problem Statement

Let

ABC

be a triangle. Point

E,F

moves on the opposite ray of

BA,CA

such that

BF=CE

. Let

M,N

be the midpoint of

BE,CF

.

BF

cuts

CE

at

D

a) Suppost that

I

is the circumcenter of

(DBE)

and

J

is the circumcenter of

(DCF)

, Prove that

MN \parallel IJ

b) Let

K

be the midpoint of

MN

and

H

be the orthocenter of triangle

AEF

. Prove that when

E

varies on the opposite ray of

BA

,

HK

go through a fixed point

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