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2015 Canada National Olympiad
2
2
Part of
2015 Canada National Olympiad
Problems
(1)
geometric inequality with altitudes! CMO 2015 P2
Source: Canadian mathematical olympiad 2015
4/24/2015
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with altitudes
A
D
,
B
E
,
AD,BE,
A
D
,
BE
,
and
C
F
CF
CF
. Let
H
H
H
be the orthocentre, that is, the point where the altitudes meet. Prove that
A
B
⋅
A
C
+
B
C
⋅
C
A
+
C
A
⋅
C
B
A
H
⋅
A
D
+
B
H
⋅
B
E
+
C
H
⋅
C
F
≤
2.
\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.
A
H
⋅
A
D
+
B
H
⋅
BE
+
C
H
⋅
CF
A
B
⋅
A
C
+
BC
⋅
C
A
+
C
A
⋅
CB
≤
2.
geometric inequality
inequalities
geometry