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Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2020 Greece JBMO TST
2
2
Part of
2020 Greece JBMO TST
Problems
(1)
sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0
Source: 2020 Greek JBMO TST p2
11/14/2020
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
1
a
+
1
b
+
1
c
=
3
\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3
a
1
+
b
1
+
c
1
=
3
. Prove that
a
+
b
a
2
+
a
b
+
b
2
+
b
+
c
b
2
+
b
c
+
c
2
+
c
+
a
c
2
+
c
a
+
a
2
≤
2
\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2
a
2
+
ab
+
b
2
a
+
b
+
b
2
+
b
c
+
c
2
b
+
c
+
c
2
+
c
a
+
a
2
c
+
a
≤
2
When is the equality valid?
algebra
inequalities