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Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2012 Ukraine Team Selection Test
1
\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab} >= 3 \sqrt{ab+bc+ca}
\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab} >= 3 \sqrt{ab+bc+ca}
Source: Ukraine TST 2012 p1
April 29, 2020
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals. Prove that
2
a
2
+
b
c
+
2
b
2
+
a
c
+
2
c
2
+
a
b
≥
3
a
b
+
b
c
+
c
a
\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}
2
a
2
+
b
c
+
2
b
2
+
a
c
+
2
c
2
+
ab
≥
3
ab
+
b
c
+
c
a
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