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Tuymaada Olympiad
2020 Tuymaada Olympiad
2
Inequality strikes again.
Inequality strikes again.
Source: Tuymaada 2020 Senior, Problem 2
October 6, 2020
Inequality
inequalities
n-variable inequality
Problem Statement
Given positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
. Let
m
=
min
(
a
1
+
1
a
2
,
a
2
+
1
a
3
,
…
,
a
n
−
1
+
1
a
n
,
a
n
+
1
a
1
)
.
m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right).
m
=
min
(
a
1
+
a
2
1
,
a
2
+
a
3
1
,
…
,
a
n
−
1
+
a
n
1
,
a
n
+
a
1
1
)
.
Prove the inequality
a
1
a
2
…
a
n
n
+
1
a
1
a
2
…
a
n
n
≥
m
.
\sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m.
n
a
1
a
2
…
a
n
+
n
a
1
a
2
…
a
n
1
≥
m
.
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