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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2007 Ukraine Team Selection Test
1
Ukraine 2007
Ukraine 2007
Source:
June 26, 2007
inequalities
inequalities proposed
Problem Statement
{
a
,
b
,
c
}
⊂
(
1
6
,
+
∞
)
\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)
{
a
,
b
,
c
}
⊂
(
6
1
,
+
∞
)
such that
a
2
+
b
2
+
c
2
=
1.
a^{2}+b^{2}+c^{2}=1.
a
2
+
b
2
+
c
2
=
1.
Prove that
1
+
a
2
2
a
2
+
3
a
b
−
c
2
+
1
+
b
2
2
b
2
+
3
b
c
−
a
2
+
1
+
c
2
2
c
2
+
3
c
a
−
b
2
≥
2
(
a
+
b
+
c
)
.
\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).
2
a
2
+
3
ab
−
c
2
1
+
a
2
+
2
b
2
+
3
b
c
−
a
2
1
+
b
2
+
2
c
2
+
3
c
a
−
b
2
1
+
c
2
≥
2
(
a
+
b
+
c
)
.
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