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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2021 Kyiv City MO
2021 Kyiv City MO Round 1
9.5
9.5
Part of
2021 Kyiv City MO Round 1
Problems
(1)
King improves upon 8.4
Source: Kyiv City MO 2021 Round 1, Problem 9.5
12/21/2023
Let
B
M
BM
BM
be the median of triangle
A
B
C
ABC
A
BC
in which
A
B
>
B
C
AB > BC
A
B
>
BC
. The point
P
P
P
is chosen so that
A
B
∥
P
C
AB\parallel PC
A
B
∥
PC
and
P
M
⊥
B
M
PM \perp BM
PM
⊥
BM
. On the line
B
P
BP
BP
, point
Q
Q
Q
is chosen so that
∠
A
Q
C
=
9
0
∘
\angle AQC = 90^\circ
∠
A
QC
=
9
0
∘
, and points
B
B
B
and
Q
Q
Q
are on opposite sides of the line
A
C
AC
A
C
. Prove that
A
B
=
B
Q
AB = BQ
A
B
=
BQ
.Proposed by Mykhailo Shtandenko
geometry