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National and Regional Contests
Romania Contests
JBMO TST - Romania
2012 Junior Balkan Team Selection Tests - Romania
1
a_{k+1} <= (a_k + a_{k+2} )/2
a_{k+1} <= (a_k + a_{k+2} )/2
Source: 2012 Romania JBMO TST 2.1
June 2, 2020
inequalities
Sequence
algebra
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be real numbers such that
a
1
=
a
n
=
a
a_1 = a_n = a
a
1
=
a
n
=
a
and
a
k
+
1
≤
a
k
+
a
k
+
2
2
a_{k+1} \le \frac{a_k + a_{k+2}}{2}
a
k
+
1
≤
2
a
k
+
a
k
+
2
, for all
k
=
1
,
2
,
.
.
.
,
n
−
2
k = 1, 2, ..., n - 2
k
=
1
,
2
,
...
,
n
−
2
. Prove that
a
k
≤
a
,
a_k \le a,
a
k
≤
a
,
for all
k
=
1
,
2
,
.
.
.
,
n
.
k = 1, 2, ..., n.
k
=
1
,
2
,
...
,
n
.
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