Olympiad problems

2012 Middle European Mathematical Olympiad, 4

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 $.

2005 National Olympiad First Round, 20

Tags: permutation statistics , inversions in permutations

We are swapping two different digits of a number in each step. If we start with the number $12345$, which of the following cannot be got after an even number of steps? $ \textbf{(A)}\ 13425 \qquad\textbf{(B)}\ 21435 \qquad\textbf{(C)}\ 35142 \qquad\textbf{(D)}\ 43125 \qquad\textbf{(E)}\ 53124 $

2000 China National Olympiad, 1

Tags: combinatorics , permutation statistics , permutation

Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “[i]innovated tuple[/i]” of $A$. The number of distinct elements in $B$ is called the “[i]innovated degree[/i]” of $A$. Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$