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Moscow Mathematical Olympiad
1940 Moscow Mathematical Olympiad
069
069
Part of
1940 Moscow Mathematical Olympiad
Problems
(1)
MMO 069 Moscow MO 1940 \frac{a_1}{a_2}+ ... + \frac{a_n}{a_1} \ge n
Source:
7/18/2019
Let
a
1
,
.
.
.
,
,
a
n
a_1, ...,, a_n
a
1
,
...
,,
a
n
be positive numbers. Prove the inequality:
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
.
.
.
+
a
n
−
1
a
n
+
a
n
a
1
≥
n
\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+ ... +\frac{a_{n-1}}{a_n}+ \frac{a_n}{a_1} \ge n
a
2
a
1
+
a
3
a
2
+
a
4
a
3
+
...
+
a
n
a
n
−
1
+
a
1
a
n
≥
n
algebra
inequalities