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Turkey Junior National Olympiad
2012 Turkey Junior National Olympiad
3
3
Part of
2012 Turkey Junior National Olympiad
Problems
(1)
A symmetric inequality with a condition
Source: Turkey Junior National Olympiad 2012 P3
12/12/2012
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers satisfying
a
3
+
b
3
+
c
3
=
a
4
+
b
4
+
c
4
a^3+b^3+c^3=a^4+b^4+c^4
a
3
+
b
3
+
c
3
=
a
4
+
b
4
+
c
4
. Show that
a
a
2
+
b
3
+
c
3
+
b
a
3
+
b
2
+
c
3
+
c
a
3
+
b
3
+
c
2
≥
1
\frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1
a
2
+
b
3
+
c
3
a
+
a
3
+
b
2
+
c
3
b
+
a
3
+
b
3
+
c
2
c
≥
1
inequalities
inequalities proposed