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Kazakhstan National Olympiad
2018 Kazakhstan National Olympiad
6
6
Part of
2018 Kazakhstan National Olympiad
Problems
(1)
Nairi Sedrakyans new geometric ineqality
Source: Kazakhstan MO 2018 final round.Grade 11;Problem 6
5/4/2018
Inside of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
found a point
M
M
M
such that
∠
A
M
B
=
∠
A
D
M
+
∠
B
C
M
\angle AMB=\angle ADM+\angle BCM
∠
A
MB
=
∠
A
D
M
+
∠
BCM
and
∠
A
M
D
=
∠
A
B
M
+
∠
D
C
M
\angle AMD=\angle ABM+\angle DCM
∠
A
M
D
=
∠
A
BM
+
∠
D
CM
.Prove that
A
M
⋅
C
M
+
B
M
⋅
D
M
≥
A
B
⋅
B
C
⋅
C
D
⋅
D
A
.
AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.
A
M
⋅
CM
+
BM
⋅
D
M
≥
A
B
⋅
BC
⋅
C
D
⋅
D
A
.
inequalities
geometric inequality
geometry
Kazakhstan