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National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2013 Saudi Arabia GMO TST
2
sum a^3/(a^2 + ab + b^2 ) >= (a + b + c)/3
sum a^3/(a^2 + ab + b^2 ) >= (a + b + c)/3
Source: 2013 Saudi Arabia GMO TST I p2
July 26, 2020
algebra
inequalities
Problem Statement
For positive real numbers
a
,
b
a, b
a
,
b
and
c
c
c
, prove that
a
3
a
2
+
a
b
+
b
2
+
b
3
b
2
+
b
c
+
c
2
+
c
3
c
2
+
c
a
+
a
2
≥
a
+
b
+
c
3
\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}
a
2
+
ab
+
b
2
a
3
+
b
2
+
b
c
+
c
2
b
3
+
c
2
+
c
a
+
a
2
c
3
≥
3
a
+
b
+
c
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