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National and Regional Contests
Romania Contests
Romania Team Selection Test
2017 Romania Team Selection Test
P3
P3
Part of
2017 Romania Team Selection Test
Problems
(1)
Maximum value of cyclic sum
Source: Romania 2017 IMO TST 2, problem 3
3/18/2018
Given an interger
n
≥
2
n\geq 2
n
≥
2
, determine the maximum value the sum
a
1
a
2
+
a
2
a
3
+
.
.
.
+
a
n
−
1
a
n
\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}
a
2
a
1
+
a
3
a
2
+
...
+
a
n
a
n
−
1
may achieve, and the points at which the maximum is achieved, as
a
1
,
a
2
,
.
.
.
a
n
a_1,a_2,...a_n
a
1
,
a
2
,
...
a
n
run over all positive real numers subject to
a
k
≥
a
1
+
a
2
.
.
.
+
a
k
−
1
a_k\geq a_1+a_2...+a_{k-1}
a
k
≥
a
1
+
a
2
...
+
a
k
−
1
, for
k
=
2
,
.
.
.
n
k=2,...n
k
=
2
,
...
n
inequalities