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2014 IMO Shortlist
G4
G4
Part of
2014 IMO Shortlist
Problems
(1)
IMO Shortlist 2014 G4
Source:
7/11/2015
Consider a fixed circle
Γ
\Gamma
Γ
with three fixed points
A
,
B
,
A, B,
A
,
B
,
and
C
C
C
on it. Also, let us fix a real number
λ
∈
(
0
,
1
)
\lambda \in(0,1)
λ
∈
(
0
,
1
)
. For a variable point
P
∉
{
A
,
B
,
C
}
P \not\in\{A, B, C\}
P
∈
{
A
,
B
,
C
}
on
Γ
\Gamma
Γ
, let
M
M
M
be the point on the segment
C
P
CP
CP
such that
C
M
=
λ
⋅
C
P
CM =\lambda\cdot CP
CM
=
λ
⋅
CP
. Let
Q
Q
Q
be the second point of intersection of the circumcircles of the triangles
A
M
P
AMP
A
MP
and
B
M
C
BMC
BMC
. Prove that as
P
P
P
varies, the point
Q
Q
Q
lies on a fixed circle.Proposed by Jack Edward Smith, UK
IMO Shortlist
geometry