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International Contests
Gulf Math Olympiad
2016 Gulf Math Olympiad
3
3
Part of
2016 Gulf Math Olympiad
Problems
(1)
triangle and orthocente
Source:
9/28/2017
Consider the acute-angled triangle
A
B
C
ABC
A
BC
. Let
X
X
X
be a point on the side
B
C
BC
BC
, and
Y
Y
Y
be a point on the side
C
A
CA
C
A
. The circle
k
1
k_1
k
1
with diameter
A
X
AX
A
X
cuts
A
C
AC
A
C
again at
E
′
E'
E
′
.The circle
k
2
k_2
k
2
with diameter
B
Y
BY
B
Y
cuts
B
C
BC
BC
again at
B
′
B'
B
′
. (i) Let
M
M
M
be the midpoint of
X
Y
XY
X
Y
. Prove that
A
′
M
=
B
′
M
A'M = B'M
A
′
M
=
B
′
M
. (ii) Suppose that
k
1
k_1
k
1
and
k
2
k_2
k
2
meet at
P
P
P
and
Q
Q
Q
. Prove that the orthocentre of
A
B
C
ABC
A
BC
lies on the line
P
Q
PQ
PQ
.
geometry