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National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1993 Turkey Team Selection Test
3
3
Part of
1993 Turkey Team Selection Test
Problems
(1)
There exists natural K such that sum > 1993/1000
Source: Turkey IMO TST 1993 #3
7/7/2011
Let (
b
n
b_n
b
n
) be a sequence such that
b
n
≥
0
b_n \geq 0
b
n
≥
0
and
b
n
+
1
2
≥
b
1
2
1
3
+
⋯
+
b
n
2
n
3
b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}
b
n
+
1
2
≥
1
3
b
1
2
+
⋯
+
n
3
b
n
2
for all
n
≥
1
n \geq 1
n
≥
1
. Prove that there exists a natural number
K
K
K
such that
∑
n
=
1
K
b
n
+
1
b
1
+
b
2
+
⋯
+
b
n
≥
1993
1000
\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}
n
=
1
∑
K
b
1
+
b
2
+
⋯
+
b
n
b
n
+
1
≥
1000
1993
inequalities unsolved
inequalities