MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN O Problems
44
44
Part of
PEN O Problems
Problems
(1)
O 44
Source:
5/25/2007
A set
C
C
C
of positive integers is called good if for every integer
k
k
k
there exist distinct
a
,
b
∈
C
a, b \in C
a
,
b
∈
C
such that the numbers
a
+
k
a+k
a
+
k
and
b
+
k
b+k
b
+
k
are not relatively prime. Prove that if the sum of the elements of a good set
C
C
C
equals
2003
2003
2003
, then there exists
c
∈
C
c \in C
c
∈
C
such that the set
C
−
{
c
}
C-\{c\}
C
−
{
c
}
is good.
modular arithmetic