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2018 International Zhautykov Olympiad
1
IZHO 2018 P1(inequality)
IZHO 2018 P1(inequality)
Source: izho 2018
February 14, 2018
algebra
inequalities
geometry
triangle inequality
Problem Statement
Let
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
measures of angles of opposite to the sides of triangle with measures
a
,
b
,
c
a,b,c
a
,
b
,
c
respectively.Prove that
2
(
c
o
s
2
α
+
c
o
s
2
β
+
c
o
s
2
γ
)
≥
a
2
b
2
+
c
2
+
b
2
a
2
+
c
2
+
c
2
a
2
+
b
2
2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}
2
(
co
s
2
α
+
co
s
2
β
+
co
s
2
γ
)
≥
b
2
+
c
2
a
2
+
a
2
+
c
2
b
2
+
a
2
+
b
2
c
2
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