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National and Regional Contests
PEN Problems
PEN N Problems
7
N 7
N 7
Source:
May 25, 2007
More Sequences
Problem Statement
Let
{
n
k
}
k
≥
1
\{n_{k}\}_{k \ge 1}
{
n
k
}
k
≥
1
be a sequence of natural numbers such that for
i
<
j
i<j
i
<
j
, the decimal representation of
n
i
n_{i}
n
i
does not occur as the leftmost digits of the decimal representation of
n
j
n_{j}
n
j
. Prove that
∑
k
=
1
∞
1
n
k
≤
1
1
+
1
2
+
⋯
+
1
9
.
\sum^{\infty}_{k=1}\frac{1}{n_{k}}\le \frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{9}.
k
=
1
∑
∞
n
k
1
≤
1
1
+
2
1
+
⋯
+
9
1
.
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