IMO Shortlist 1969 problem 25
Dodao/la:
arhiva2. travnja 2012. ![(GBR 2)](/media/m/1/8/7/187b52a51892589879f9ab1b7a067251.png)
Let
![a, b, x, y](/media/m/d/7/f/d7fac2b7af37bbbc219671d052013bc9.png)
be positive integers such that
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
have no common divisor greater than
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
. Prove that the largest number not expressible in the form
![ax + by](/media/m/c/3/0/c30aa7ca4db0141672aba6e6660a78f9.png)
is
![ab - a - b](/media/m/6/b/a/6ba09ebb93109bf5d91798f2a6fdf12a.png)
. If
![N(k)](/media/m/3/b/e/3be846f12a141464d7fdbbbe72bb564f.png)
is the largest number not expressible in the form
![ax + by](/media/m/c/3/0/c30aa7ca4db0141672aba6e6660a78f9.png)
in only
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
ways, find
%V0
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
Izvor: Međunarodna matematička olimpijada, shortlist 1969