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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2010 Ukraine Team Selection Test
4
a/(b+c)+b/(c+a)+c/(a+b)+\sqrt{(ab+bc+ca)/(a^2+b^2+c^2)}>= 5/2
a/(b+c)+b/(c+a)+c/(a+b)+\sqrt{(ab+bc+ca)/(a^2+b^2+c^2)}>= 5/2
Source: Ukraine TSt 2010 p4
May 4, 2020
inequalities
algebra
Problem Statement
For the nonnegative numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
prove the inequality:
a
b
+
c
+
b
c
+
a
+
c
a
+
b
+
a
b
+
b
c
+
c
a
a
2
+
b
2
+
c
2
≥
5
2
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52
b
+
c
a
+
c
+
a
b
+
a
+
b
c
+
a
2
+
b
2
+
c
2
ab
+
b
c
+
c
a
≥
2
5
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