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.

A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.

A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.

Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

A permutation of the integers \(2020, 2021,...,2118, 2119\) is a list \(a_1,a_2,a_3,...,a_{100}\) where each one of the numbers appears exactly once. For each permutation we define the partial sums. $s_1=a_1$ $s_2=a_1+a_2$ $s_3=a_1+a_2+a_3$ $...$ $s_{100}=a_1+a_2+...+a_{100}$ How many of these permutations satisfy that none of the numbers \(s_1,...,s_{100}\) is divisible by $3$?

Let $x_1,x_2,...,x_k$ be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a [b]scramble[/b] if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is $1,3,3,5$ then $3,5,1,3$ is a scramble, but $3,3,1,5$ is not. A rearrangement is called a [b]two-two[/b] if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3,5,1,3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$). Let $n\geq 2$. Prove that the number of scrambles of $1,1,2,3,...,n-1,n$ is equal to the number of two-twos of $1,2,3,...,n,n+1$. (Notice that both sequences have $n+1$ numbers, but the first one contains two 1s.)

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

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 both of their neighbours remained and exactly one third of the dwarves have only one of their neighbours remained, if the number of the dwarves is a) 6; b) 9?

A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal? [i]Proposed by Mariia Rozhkova[/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 $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

Let $a_1, a_2, ...,a_n$ represent an arbitrary arrangement of the numbers $1, 2, ...,n$. Prove that, if $n$ is odd, the product $$(a_1 - 1)(a_2 - 2) ... (a_n -n)$$ is an even number.

Does there exist a sequence of positive integers \( a_1, a_2, \ldots, a_{100} \) such that every number from \( 1 \) to \( 100 \) appears exactly once, and for each \( 1 \leq i \leq 100 \), the condition \[ a_{a_i + i} = i \] holds? Here it is assumed that \( a_{k+100} = a_k \) for each \( 1 \leq k \leq 100 \). [i]Proposed by Mykhailo Shtandenko[/i]

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 $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.

Two permutations of $1,\ldots, n$ are written on the board: $a_1,\ldots,a_n$ $b_1,\ldots,b_n$ A move consists of one of the following two operations: 1) Change the first row to $b_{a_1},\ldots,b_{a_n}$ 2) Change the second row to $a_{b_1},\ldots,a_{b_n}$ The starting position is: $2134\ldots n$ $234\ldots n1$ Is it possible by finitely many moves to get: $2314\ldots n$ $234 \ldots n1$? [i]D. Zmiaikou[/i]

1) Prove that there are infinitely many positive integers $n$ such that there exists a permutation of $1, 2, 3, . . . , n$ with the property that the difference between any two adjacent numbers is equal to either $2015$ or $2016$. 2) Let $k$ be a positive integer. Is the statement in 1) still true if we replace the numbers $2015$ and $2016$ by $k$ and $k + 2016$, respectively?

Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.

From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.

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 $a_1 , a_2 ,... , a_n$ be an arrangement of the integers $1,2,..., n$. Let $$S=\frac{a_1}{1}+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{1}.$$ Find a natural number $n$ such that among the values of $S$ for all arrangements $a_1 , a_2 ,... , a_n$ , all the integers from $n$ to $n + 100$ appear .

$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$

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.

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 ?