IMO Shortlist 1967 problem 1
Dodao/la:
arhiva2. travnja 2012. Let
![k,m,n](/media/m/e/b/0/eb02bdc155b91d7a4e7108860107a69d.png)
be natural numbers such that
![m+k+1](/media/m/8/c/c/8cce1ef5480d15679a6dfed9a68aaec9.png)
is a prime greater than
![n+1](/media/m/2/a/7/2a7327e09a84d01a602088c9f045cbde.png)
. Let
![c_s=s(s+1)](/media/m/8/5/b/85b9f2bbdb515ea8a8678cd608f427fa.png)
. Prove that
is divisible by the product
![c_1c_2\ldots c_n](/media/m/0/c/9/0c9eff3992151305b9addc110a1eb5c8.png)
.
%V0
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that
$$(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)$$
is divisible by the product $c_1c_2\ldots c_n$.
Izvor: Međunarodna matematička olimpijada, shortlist 1967