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Ireland National Math Olympiad
1992 Irish Math Olympiad
5
nth Root Inequality
nth Root Inequality
Source: 1992 IrMO Paper 2 Problem 5
October 2, 2017
Inequality
inequalities
Problem Statement
If, for
k
=
1
,
2
,
…
,
n
k=1,2,\dots ,n
k
=
1
,
2
,
…
,
n
,
a
k
a_k
a
k
and
b
k
b_k
b
k
are positive real numbers, prove that
a
1
a
2
⋯
a
n
n
+
b
1
b
2
⋯
b
n
n
≤
(
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
n
;
\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};
n
a
1
a
2
⋯
a
n
+
n
b
1
b
2
⋯
b
n
≤
n
(
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
;
and that equality holds if, and only if,
a
1
b
1
=
a
2
b
2
=
⋯
=
a
n
b
n
.
\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.
b
1
a
1
=
b
2
a
2
=
⋯
=
b
n
a
n
.
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