IMO Shortlist 1987 problem 18
Dodao/la:
arhiva2. travnja 2012. For any integer
![r \geq 1](/media/m/2/7/2/27280ad9f9411994c4705ae3f202f9a8.png)
, determine the smallest integer
![h(r) \geq 1](/media/m/6/0/8/60809267b2e7b8062d852b5460f6df73.png)
such that for any partition of the set
![\{1, 2, \cdots, h(r)\}](/media/m/6/6/2/6623157f661c8ebca9485d8597d4387b.png)
into
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
classes, there are integers
![a \geq 0 \ ; 1 \leq x \leq y](/media/m/2/9/c/29c8e45d5125280f0fa5bba827d27a8a.png)
, such that
![a + x, a + y, a + x + y](/media/m/4/c/1/4c1ac30288fcd1e5307ab7bbc8cba1c2.png)
belong to the same class.
Proposed by Romania
%V0
For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.
Proposed by Romania
Izvor: Međunarodna matematička olimpijada, shortlist 1987