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National and Regional Contests
Azerbaijan Contests
Azerbaijan Team Selection Test
2022 Azerbaijan IMO TST
2022 Azerbaijan IMO TST
Part of
Azerbaijan Team Selection Test
Subcontests
(1)
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Random point on circumcircle makes a equal angle made with reflections
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
ω
\omega
ω
and
D
D
D
be any point on
ω
.
\omega.
ω
.
Suppose that
P
P
P
is the midpoint of chord
A
D
AD
A
D
and points
X
,
Y
X, Y
X
,
Y
are chosen on lines
A
C
,
A
B
AC, AB
A
C
,
A
B
such that reflections of
B
,
C
B, C
B
,
C
with respect to
A
D
AD
A
D
lie on
X
P
,
Y
P
,
XP, YP,
XP
,
Y
P
,
respectively. If the circumcircle of triangle
A
X
Y
AXY
A
X
Y
intersects
ω
\omega
ω
at
I
I
I
for the second time, prove that
∠
P
I
D
\angle PID
∠
P
I
D
equals the angle formed by lines
A
D
AD
A
D
and
B
C
.
BC.
BC
.
Proposed by tenplusten.