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Czech-Polish-Slovak Match
2023 Czech-Polish-Slovak Match
3
3
Part of
2023 Czech-Polish-Slovak Match
Problems
(1)
Weird angle conditions again
Source: CAPS Match 2023 P3
6/30/2023
Given is a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
∠
B
A
D
=
∠
B
C
D
\angle BAD = \angle BCD
∠
B
A
D
=
∠
BC
D
and
∠
A
B
C
<
∠
A
D
C
\angle ABC < \angle ADC
∠
A
BC
<
∠
A
D
C
. Point
M
M
M
is the midpoint of segment
A
C
AC
A
C
. Prove that there exist points
X
X
X
and
Y
Y
Y
on the segments
A
B
AB
A
B
and
B
C
BC
BC
, respectively, such that
X
Y
⊥
B
D
,
M
X
=
M
Y
XY \perp BD, MX = MY
X
Y
⊥
B
D
,
MX
=
M
Y
and
∠
X
M
Y
=
∠
A
D
C
−
∠
A
B
C
\angle XMY = \angle ADC - \angle ABC
∠
XM
Y
=
∠
A
D
C
−
∠
A
BC
.Proposed by Mykhailo Shtandenko