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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia and Herzegovina BMO TST
2022 Bosnia and Herzegovina BMO TST
1
1
Part of
2022 Bosnia and Herzegovina BMO TST
Problems
(1)
Bosnia and Herzegovina 2022 BMO TST P1
Source:
5/22/2022
Let
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3, \ldots
a
1
,
a
2
,
a
3
,
…
be an infinite sequence of nonnegative real numbers such that for all positive integers
k
k
k
the following conditions hold:
i
)
i)
i
)
a
k
−
2
a
k
+
1
+
a
k
+
2
≥
0
a_k-2a_{k+1}+a_{k+2} \geq 0
a
k
−
2
a
k
+
1
+
a
k
+
2
≥
0
;
i
i
)
ii)
ii
)
∑
j
=
1
k
a
j
≤
1
\sum_{j=1}^{k} a_j \leq 1
∑
j
=
1
k
a
j
≤
1
. Prove that for all positive integer
k
k
k
holds:
0
≤
a
k
−
a
k
+
1
<
2
k
2
0 \leq a_k - a_{k+1} < \frac{2}{k^2}
0
≤
a
k
−
a
k
+
1
<
k
2
2
Sequence
algebra