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China South East Mathematical Olympiad 2021 Grade11 Q3
China South East Mathematical Olympiad 2021 Grade11 Q3
Source: China Nanchang, Jul 28, 2021
July 28, 2021
inequalities
algebra
China
Problem Statement
Let
a
,
b
,
c
≥
0
a,b,c\geq 0
a
,
b
,
c
≥
0
and
a
2
+
b
2
+
c
2
≤
1.
a^2+b^2+c^2\leq 1.
a
2
+
b
2
+
c
2
≤
1.
Prove that
a
a
2
+
b
c
+
1
+
b
b
2
+
c
a
+
1
+
c
c
2
+
a
b
+
1
+
3
a
b
c
<
3
\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3
a
2
+
b
c
+
1
a
+
b
2
+
c
a
+
1
b
+
c
2
+
ab
+
1
c
+
3
ab
c
<
3
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