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239 Open Math Olympiad
2013 239 Open Mathematical Olympiad
7
Geometric inequality in convex quadrilateral
Geometric inequality in convex quadrilateral
Source: 239 2013 S7
August 7, 2020
geometry
Geometric Inequalities
Problem Statement
Point
M
M
M
is the midpoint of side
B
C
BC
BC
of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. If
∠
A
M
D
<
12
0
∘
\angle{AMD} < 120^{\circ}
∠
A
M
D
<
12
0
∘
. Prove that
(
A
B
+
A
M
)
2
+
(
C
D
+
D
M
)
2
>
A
D
⋅
B
C
+
2
A
B
⋅
C
D
.
(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD.
(
A
B
+
A
M
)
2
+
(
C
D
+
D
M
)
2
>
A
D
⋅
BC
+
2
A
B
⋅
C
D
.
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