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National and Regional Contests
Poland Contests
Poland - Second Round
1967 Poland - Second Round
1
1
Part of
1967 Poland - Second Round
Problems
(1)
a_{k-1}+a_{k+1} >= 2a_k
Source: Polish MO second round 1967 p1
8/22/2024
Real numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
(
n
≥
3
n \ge 3
n
≥
3
) satisfy the conditions
a
1
=
a
n
=
0
a_1 = a_n = 0
a
1
=
a
n
=
0
and
a
k
−
1
+
a
k
+
1
≥
2
a
k
a_{k-1}+a_{k+1} \ge 2a_k
a
k
−
1
+
a
k
+
1
≥
2
a
k
for
k
=
2
k = 2
k
=
2
,
3
3
3
,
.
.
.
,
,...,
,
...
,
n
−
1
n -1
n
−
1
. Prove that none of the numbers
a
1
a_1
a
1
,
.
.
.
...
...
,
a
n
a_n
a
n
is positive.
algebra
inequalities