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Baltic Way
2014 Baltic Way
2
Sequence with non-positive terms
Sequence with non-positive terms
Source: Baltic Way 2014, Problem 2
November 11, 2014
algebra proposed
algebra
Problem Statement
Let
a
0
,
a
1
,
.
.
.
,
a
N
a_0, a_1, . . . , a_N
a
0
,
a
1
,
...
,
a
N
be real numbers satisfying
a
0
=
a
N
=
0
a_0 = a_N = 0
a
0
=
a
N
=
0
and
a
i
+
1
−
2
a
i
+
a
i
−
1
=
a
i
2
a_{i+1} - 2a_i + a_{i-1} = a^2_i
a
i
+
1
−
2
a
i
+
a
i
−
1
=
a
i
2
for
i
=
1
,
2
,
.
.
.
,
N
−
1.
i = 1, 2, . . . , N - 1.
i
=
1
,
2
,
...
,
N
−
1.
Prove that
a
i
≤
0
a_i\leq 0
a
i
≤
0
for
i
=
1
,
2
,
.
.
.
,
N
−
1.
i = 1, 2, . . . , N- 1.
i
=
1
,
2
,
...
,
N
−
1.
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