Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order. We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.

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 $M_n$ be the set of permutations $\sigma\in S_n$ for which there exists $\tau\in S_n$ such that the numbers \[\sigma (1)+\tau(1),\, \sigma(2)+\tau(2),\ldots,\sigma(n)+\tau(n),\] are consecutive. Show that \((M_n\neq \emptyset\Leftrightarrow n\text{ is odd})\) and in this case for each $\sigma_1,\sigma_2\in M_n$ the following equality holds: \[\sum_{k=1}^n k\sigma_1(k)=\sum_{k=1}^n k\sigma_2(k).\] [i]Dan Schwarz[/i]

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$. Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$, we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling. (a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line? (b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?

Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have two new neighbours and exactly one third of the dwarves have only one new neighbour, if the number of the dwarves is a) 9; b) 12?

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]

Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that (a) The sum of the digits of $2M$ equals the sum of the digits of $2K$. (b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even). (c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$. (AD Lisitskiy)

Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence. Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.

Determine all natural numbers $n \ge 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers. [i](Walther Janous)[/i]

Let $n\geqslant 3$ be an integer and $a_1,a_2,\ldots,a_n$ be pairwise distinct positive real numbers with the property that there exists a permutation $b_1,b_2,\ldots,b_n$ of these numbers such that\[\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_{n-1}}{b_{n-1}}\neq 1.\]Prove that there exist $a,b>0$ such that $\{a_1,a_2,\ldots,a_n\}=\{ab,ab^2,\ldots,ab^n\}.$ [i]Cristi Săvescu[/i]

For positive integers $k$ and $n$, express the number of permutation $P=x_1x_2...x_{2n}$ consisting of $A$ and $B$ that satisfies all three of the following conditions, using $k$ and $n$. $ $ $ $ $(i)$ $A, B$ appear exactly $n$ times respectively in $P$. $ $ $ $ $(ii)$ For each $1\le i\le n$, if we denote the number of $A$ in $x_1,x_2,...,x_i$ as $a_i$ $,$ then $\mid 2a_i -i\mid \le 1$. $ $ $ $ $(iii)$ $AB$ appears exactly $k$ times in $P$. (For example, $AB$ appears 3 times in $ABBABAAB$)

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different. Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

The PMO Magician has a special party game. There are $n$ chairs, labelled $1$ to $n$. There are $n$ sheets of paper, labelled $1$ to $n$. [list] [*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks $n$ party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number. [/list] Show that if $1 < m \leq n$, where $m$ is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly $m$ claps.

In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that: *$\sigma (i) \neq i$ for all $i=1,2,...,n$ * there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $ Determine the limit $\lim_{n \to + \infty} \frac{f(n)}{n!}$

Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$. Evaluate $a_{n,k}-b_{n,k}$. [i]* * *[/i]

Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products? [i](C. Wu)[/i]

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.

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 $2010$.