IMO Shortlist 1981 problem 8
Dodao/la:
arhiva2. travnja 2012. Take
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
such that
![1\le r\le n](/media/m/f/b/b/fbb5395e82e42b2b081e2c0404bf1d88.png)
, and consider all subsets of
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
elements of the set
![\{1,2,\ldots,n\}](/media/m/9/6/a/96a5b5240166685da05abd76f5be8328.png)
. Each subset has a smallest element. Let
![F(n,r)](/media/m/a/a/0/aa04a5d0084a4fa1235035596e6f1530.png)
be the arithmetic mean of these smallest elements. Prove that:
%V0
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: $$F(n,r)={n+1\over r+1}.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1981