We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

Denote every permutation of $1,2,\dots, n$ as $\sigma =(a_1,a_2,\dots,n)$. Prove that the sum $$\sum \frac{1}{(a_1)(a_1+a_2)(a_1+a_2+a_3)\dots(a_1+a_2+\dots+a_n)}$$ taken over all possible permutations $\sigma$ equals $\frac{1}{n!}$.

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

Given a positive integer $n$, determine how many permutations $\sigma$ of the set $\{1, 2, \ldots , 2022n\}$ have the following property: for each $i \in \{1, 2, \ldots , 2021n + 1\}$, the number $$\sigma(i) + \sigma(i + 1) + \cdots + \sigma(i + n - 1)$$ is a multiple of $n$.

Consider the string of length $6$ composed of three characters $a, b, c$. For each string, if two $a$s are next to each other, or two $b$s are next to each other, then replace $aa$ by $b$, and replace $bb$ by $a$. Also, if $a$ and $b$ are next to each other, or two $c$s are next to each other, remove all two of them (i.e. delete $ab, ba, cc$). Determine the number of strings that can be reduced to $c$, the string of length $1$, by the reducing processes mentioned above.

Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

In the following exercise, $ C_G (e) $ denotes the centralizer of the element $ e $ in the group $ G. $ [b]a)[/b] Prove that $ \max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right| <\frac{n!}{2} , $ for any natural number $ n\ge 4. $ [b]b)[/b] Show that $ \lim_{n\to\infty} \left(\frac{1}{n!}\cdot\max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right|\right) =0. $ [i]Alexandru Cioba[/i]

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

A permutation of the integers $1, 2, \ldots, m$ is called [i]fresh[/i] if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$. Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$. [i]For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.[/i]

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.

Let $p$ and $q$ be mutually prime natural numbers greater than $1$. Starting with the permutation $(1, 2, . . . , n)$, in one move we can switch the places of two numbers if their difference is $p$ or $q$. Prove that with such moves we can get any another permutation if and only if $n \ge p + q - 1$.

Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.

We consider an integer $n \ge 3,$ the set $S=\{1,2,3,\ldots,n\}$ and the set $\mathcal{F}$ of the functions from $S$ to $S.$ We say that $\mathcal{G} \subset \mathcal{F}$ is a generating set for $\mathcal{H} \subset \mathcal{F}$ if any function in $\mathcal{H}$ can be represented as a composition of functions from $\mathcal{G}.$ a) Let the functions $a:S \to S,$ $a(n-1)=n,$ $a(n)=n-1$ and $a(k)=k$ for $k \in S \setminus \{n-1,n\}$ and $b:S \to S,$ $b(n)=1$ and $b(k)=k+1$ for $k \in S \setminus \{n\}.$ Prove that $\{a,b\}$ is a generating set for the set $\mathcal{B}$ of bijective functions of $\mathcal{F}.$ b) Prove that the smallest number of elements that a generating set of $\mathcal{F}$ has is $3.$

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which \[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that : $$p(p(i)) \geq i$$

$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is [i][u]expanding[/u][/i] for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$ $\indent$ [b](a)[/b] Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ? $\indent$ [b](b)[/b] Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?

In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]