MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN O Problems
25
25
Part of
PEN O Problems
Problems
(1)
O 25
Source:
5/25/2007
Let
A
A
A
be a non-empty set of positive integers. Suppose that there are positive integers
b
1
b_{1}
b
1
,
⋯
\cdots
⋯
,
b
n
b_{n}
b
n
and
c
1
c_{1}
c
1
,
⋯
\cdots
⋯
,
c
n
c_{n}
c
n
such that [*] for each
i
i
i
the set
b
i
A
+
c
i
=
{
b
i
a
+
c
i
∣
a
∈
A
}
b_{i}A+c_{i}=\{b_{i}a+c_{i}\vert a \in A \}
b
i
A
+
c
i
=
{
b
i
a
+
c
i
∣
a
∈
A
}
is a subset of
A
A
A
, [*] the sets
b
i
A
+
c
i
b_{i}A+c_{i}
b
i
A
+
c
i
and
b
j
A
+
c
j
b_{j}A+c_{j}
b
j
A
+
c
j
are disjoint whenever
i
≠
j
i \neq j
i
=
j
. Prove that
1
b
1
+
⋯
+
1
b
n
≤
1.
\frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\le 1.
b
1
1
+
⋯
+
b
n
1
≤
1.
limit