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National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
9.5
9.5
Part of
2023 All-Russian Olympiad Regional Round
Problems
(1)
Tangent circles in a quadrilateral
Source: ARO Regional stage 2023 9.5
2/16/2023
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that the circles with diameters
A
B
AB
A
B
and
C
D
CD
C
D
touch at
S
S
S
. If
M
,
N
M, N
M
,
N
are the midpoints of
A
B
,
C
D
AB, CD
A
B
,
C
D
, prove that the perpendicular through
M
M
M
to
M
N
MN
MN
meets
C
S
CS
CS
on the circumcircle of
A
B
C
D
ABCD
A
BC
D
.
geometry
Russia