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National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2023 Kazakhstan National Olympiad
3
Inequality with some condition
Inequality with some condition
Source: Kazakhstan National Olympiad 2023/ Problem 3
March 21, 2023
inequalities
Problem Statement
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
max
{
a
(
b
+
c
)
a
2
+
b
c
,
b
(
c
+
a
)
b
2
+
c
a
,
c
(
a
+
b
)
c
2
+
a
b
}
≤
5
2
\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}
max
{
a
2
+
b
c
a
(
b
+
c
)
,
b
2
+
c
a
b
(
c
+
a
)
,
c
2
+
ab
c
(
a
+
b
)
}
≤
2
5
. Prove inequality
a
(
b
+
c
)
a
2
+
b
c
+
b
(
c
+
a
)
b
2
+
c
a
+
c
(
a
+
b
)
c
2
+
a
b
≤
3
\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3
a
2
+
b
c
a
(
b
+
c
)
+
b
2
+
c
a
b
(
c
+
a
)
+
c
2
+
ab
c
(
a
+
b
)
≤
3
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