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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2021 Ukraine National Mathematical Olympiad
9.4
9.4
Part of
2021 Ukraine National Mathematical Olympiad
Problems
(1)
Triangle with 60° angle, projection of othocenter onto median
Source: Ukraine MO 2021 9.4
5/2/2021
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
∘
\angle A = 60^\circ
∠
A
=
6
0
∘
. Points
P
P
P
and
Q
Q
Q
are chosen on sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
P
=
P
Q
=
Q
C
BP = PQ = QC
BP
=
PQ
=
QC
. Prove that the circumcircle of
△
A
P
Q
\triangle APQ
△
A
PQ
passes through the projection of the orthocenter of
A
B
C
ABC
A
BC
onto its
A
A
A
-median.Proposed by Fedir Yudin
geometry
circumcircle