2006 Romanian NMO Problems/Grade 8/Problem 2
Problem
Let be a positive integer. Prove that there exists an integer
,
, and numbers
, such that
![$n = \sum_{1\leq i < j \leq k } a_ia_j$](http://latex.artofproblemsolving.com/c/3/f/c3f8748bf68501f3212d2a2fabd83d374f793ea2.png)
Solution
Suppose that there exists such an integer and numbers
,
such that
. Let
then
We see that if
and
are increased by
, then
is decreased by
. This motivates the following construction: Let
be large enough such that
, let
and
, let
, and let
and
. Then