MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2018 ITAMO
2
2
Part of
2018 ITAMO
Problems
(1)
ITAMO 2018 problem 2
Source:
5/9/2018
2.
2.
2.
Let
A
B
C
ABC
A
BC
be an acute-angeled triangle , non-isosceles and with barycentre
G
G
G
(which is , in fact , the intersection of the medians).Let
M
M
M
be the midpoint of
B
C
BC
BC
, and let Ω be the circle with centre
G
G
G
and radius
G
M
GM
GM
, and let
N
N
N
be the point of intersection between Ω and
B
C
BC
BC
that is distinct from
M
M
M
.Let
S
S
S
be the symmetric point of
A
A
A
with respect to
N
N
N
, that is , the point on the line
A
N
AN
A
N
such that
A
N
=
N
S
AN=NS
A
N
=
NS
.Prove that
G
S
GS
GS
is perpendicular to
B
C
BC
BC
geometry