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National and Regional Contests
Russia Contests
239 Open Math Olympiad
2012 239 Open Mathematical Olympiad
4
Four variable inequality
Four variable inequality
Source: 239 2012 S4
July 30, 2020
inequalities
algebra
Problem Statement
For some positive numbers
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
, we know that
1
a
3
+
1
+
1
b
3
+
1
+
1
c
3
+
1
+
1
d
3
+
1
=
2.
\frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2.
a
3
+
1
1
+
b
3
+
1
1
+
c
3
+
1
1
+
d
3
+
1
1
=
2.
Prove that
1
−
a
a
2
−
a
+
1
+
1
−
b
b
2
−
b
+
1
+
1
−
c
c
2
−
c
+
1
+
1
−
d
d
2
−
d
+
1
≥
0.
\frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0.
a
2
−
a
+
1
1
−
a
+
b
2
−
b
+
1
1
−
b
+
c
2
−
c
+
1
1
−
c
+
d
2
−
d
+
1
1
−
d
≥
0.
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