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2010 IMC
5
IMC 2010 - Problem 5
IMC 2010 - Problem 5
Source:
July 26, 2010
inequalities unsolved
inequalities
Problem Statement
Suppose that
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers in the interval
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
such that
1
+
2
a
b
c
≥
a
2
+
b
2
+
c
2
1 + 2abc \geq a^2+b^2+c^2
1
+
2
ab
c
≥
a
2
+
b
2
+
c
2
. Prove that
1
+
2
(
a
b
c
)
n
≥
a
2
n
+
b
2
n
+
c
2
n
1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n}
1
+
2
(
ab
c
)
n
≥
a
2
n
+
b
2
n
+
c
2
n
for all positive integers
n
n
n
.
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