MEMO 2012 ekipno problem 4
Dodao/la:
arhiva23. lipnja 2013. Let
![p>2](/media/m/7/4/c/74cac2be359b29114e872b34bebfb11b.png)
be a prime number. For any permutation
![\pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) )](/media/m/7/f/e/7fea2b5cdc6b7053f9abfd73396ecdf4.png)
of the set
![S = \{ 1, 2, \cdots , p \}](/media/m/6/8/4/684e085c31df6aefa9865e901e5d6735.png)
, let
![f( \pi )](/media/m/a/2/d/a2da695f93a99e858b994edb966bbde2.png)
denote the number of multiples of
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
among the following
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
numbers:
![\pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)](/media/m/e/c/a/ecacf2788a177944d91a35e45fe05065.png)
Determine the average value of
![f( \pi)](/media/m/1/f/f/1ff914260ed03f7670b390071e4a2730.png)
taken over all permutations
![\pi](/media/m/6/d/c/6dc45296009278a7c7756c5f81a379fb.png)
of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
.
%V0
Let $p>2$ be a prime number. For any permutation $\pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) )$ of the set $S = \{ 1, 2, \cdots , p \}$, let $f( \pi )$ denote the number of multiples of $p$ among the following $p$ numbers:
$$\pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)$$
Determine the average value of $f( \pi)$ taken over all permutations $\pi$ of $S$.
Izvor: Srednjoeuropska matematička olimpijada 2012, ekipno natjecanje, problem 4