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Polish MO Finals
1975 Polish MO Finals
1
\sum_{i=N}^{N+k} a_i \ge 0
\sum_{i=N}^{N+k} a_i \ge 0
Source: Polish MO Finals 1975 p1
August 23, 2024
algebra
inequalities
Problem Statement
A sequence
(
a
k
)
k
=
1
∞
(a_k)_{k=1}^{\infty}
(
a
k
)
k
=
1
∞
has the property that there is a natural number
n
n
n
such that
a
1
+
a
2
+
.
.
.
+
a
n
=
0
a_1 + a_2 +...+ a_n = 0
a
1
+
a
2
+
...
+
a
n
=
0
and
a
n
+
k
=
a
k
a_{n+k} = a_k
a
n
+
k
=
a
k
for all
k
k
k
. Prove that there exists a natural number
N
N
N
such that
∑
i
=
N
N
+
k
a
i
≥
0
f
o
r
k
=
0
,
1
,
2...
\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...
i
=
N
∑
N
+
k
a
i
≥
0
f
or
k
=
0
,
1
,
2...
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