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India IMO Training Camp
2010 India IMO Training Camp
4
Indian Team Selection Test 2010 ST2 P1
Indian Team Selection Test 2010 ST2 P1
Source:
May 22, 2010
inequalities
function
IMO Shortlist
inequalities unsolved
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
+
b
c
+
c
a
≤
3
a
b
c
ab+bc+ca\le 3abc
ab
+
b
c
+
c
a
≤
3
ab
c
. Prove that
a
2
+
b
2
a
+
b
+
b
2
+
c
2
b
+
c
+
c
2
+
a
2
c
+
a
+
3
≤
2
(
a
+
b
+
b
+
c
+
c
+
a
)
\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})
a
+
b
a
2
+
b
2
+
b
+
c
b
2
+
c
2
+
c
+
a
c
2
+
a
2
+
3
≤
2
(
a
+
b
+
b
+
c
+
c
+
a
)
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