MathDB
Problems
Contests
International Contests
IMO Longlists
1988 IMO Longlists
5
5
Part of
1988 IMO Longlists
Problems
(1)
Partition M_k into two subsets
Source: IMO LongList 1988, Cuba 1, Problem 5 of ILL
10/22/2005
Let
k
k
k
be a positive integer and
M
k
M_k
M
k
the set of all the integers that are between
2
⋅
k
2
+
k
2 \cdot k^2 + k
2
⋅
k
2
+
k
and
2
⋅
k
2
+
3
⋅
k
,
2 \cdot k^2 + 3 \cdot k,
2
⋅
k
2
+
3
⋅
k
,
both included. Is it possible to partition
M
k
M_k
M
k
into 2 subsets
A
A
A
and
B
B
B
such that
∑
x
∈
A
x
2
=
∑
x
∈
B
x
2
.
\sum_{x \in A} x^2 = \sum_{x \in B} x^2.
x
∈
A
∑
x
2
=
x
∈
B
∑
x
2
.
combinatorics unsolved
combinatorics