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International Contests
Mediterranean Mathematics Olympiad
2000 Mediterranean Mathematics Olympiad
4
Ineq
Ineq
Source: Mediterranean MO 2000
August 26, 2005
vector
geometry solved
geometry
Problem Statement
Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
be the midpoints of the sides
B
C
,
C
D
,
D
A
,
A
B
BC,CD,DA,AB
BC
,
C
D
,
D
A
,
A
B
of a convex quadrilateral, respectively. Prove that
4
(
A
P
2
+
B
Q
2
+
C
R
2
+
D
S
2
)
≤
5
(
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
)
4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)
4
(
A
P
2
+
B
Q
2
+
C
R
2
+
D
S
2
)
≤
5
(
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
)
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