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National and Regional Contests
Serbia Contests
Serbia JBMO TST
2019 Serbia JBMO TST
3
3
Part of
2019 Serbia JBMO TST
Problems
(1)
JBMO TST Serbia Problem 3
Source:
7/7/2019
3.
3.
3.
Congruent circles
k
1
k_{1}
k
1
and
k
2
k_{2}
k
2
intersect in the points
A
A
A
and
B
B
B
. Let
P
P
P
be a variable point of arc
A
B
AB
A
B
of circle
k
2
k_{2}
k
2
which is inside
k
1
k_{1}
k
1
and let
A
P
AP
A
P
intersect
k
1
k_{1}
k
1
once more in point
C
C
C
, and the ray
C
B
CB
CB
intersects
k
2
k_{2}
k
2
once more in
D
D
D
. Let the angle bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
intersect
k
1
k_{1}
k
1
in
E
E
E
, and the circle
k
2
k_{2}
k
2
in
F
F
F
. Ray
F
B
FB
FB
intersects
k
1
k_{1}
k
1
in
Q
Q
Q
. If
X
X
X
is one of the intersection points of circumscribed circles of triangles
C
D
P
CDP
C
D
P
and
E
Q
F
EQF
EQF
, prove that the triangle
C
F
X
CFX
CFX
is equilateral.
geometry