MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2001 Moldova Team Selection Test
12
12
Part of
2001 Moldova Team Selection Test
Problems
(1)
$2k$ divides $n(n+1)$ and $2k\leq n+1$
Source: Moldova TST 2001
8/7/2023
Let
n
n{}
n
(
n
≥
1
)
(n\geq 1)
(
n
≥
1
)
be an integer and a set
A
=
{
1
,
2
,
…
,
n
}
A=\{1,2,\ldots,n\}
A
=
{
1
,
2
,
…
,
n
}
. The set
A
A{}
A
is
k
−
p
a
r
t
i
t
i
o
n
a
b
l
e
k-partitionable
k
−
p
a
r
t
i
t
i
o
nab
l
e
if it can be partitioned in
k
k{}
k
disjoint sets with the same sum of elements. Show that
A
A{}
A
is
k
−
p
a
r
t
i
t
i
o
n
a
b
l
e
k-partitionable
k
−
p
a
r
t
i
t
i
o
nab
l
e
if and only if
2
k
2k
2
k
divides
n
(
n
+
1
)
n(n+1)
n
(
n
+
1
)
and
2
k
≤
n
+
1
2k\leq n+1
2
k
≤
n
+
1
.
number theory