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Ersatz MO (USEMO)
2024 USEMO
5
5
Part of
2024 USEMO
Problems
(1)
Tangency to EFG
Source: USEMO 2024/5
10/27/2024
Let
A
B
C
ABC
A
BC
be a scalene triangle whose incircle is tangent to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
respectively. Lines
B
E
BE
BE
and
C
F
CF
CF
meet at
G
G
G
. Prove that there exists a point
X
X
X
on the circumcircle of triangle
E
F
G
EFG
EFG
such that the circumcircles of triangles
B
C
X
BCX
BCX
and
E
F
G
EFG
EFG
are tangent, and
∠
B
G
C
=
∠
B
X
C
+
∠
E
D
F
.
\angle BGC = \angle BXC + \angle EDF.
∠
BGC
=
∠
BXC
+
∠
E
D
F
.
Kornpholkrit Weraarchakul
geometry
USEMO 2024