For a permutation $p(a_1,a_2,...,a_{17})$ of $1,2,...,17$, let $k_p$ denote the largest $k$ for which $a_1 +...+a_k < a_{k+1} +...+a_{17}$. Find the maximum and minimum values of $k_p$ and find the sum $\sum_{p} k_p$ over all permutations$ p$.

An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$

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)

Consider a table of cyclic permutations ($n\ge2$) \[ \begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix} \] Then multiply each number of the first row by that number of the $k$-th row that is in the same column. Sum all these products and denote $s_k$ the result (e.g. $s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1$). a) Find a recursive relation for $s_k$ in terms of $s_{k-1}$ and determine the explicit formula for $s_k$. b) Determine both an index $k$ and the value of $s_k$ such that the sum $s_k$ is minimal.

A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers $b_1=a_1$ $b_2= a_1 + a_2$ $b_3=a_1 + a_2 + a_3$ ... $b_{100}=a_1 + a_2 + ...+a_{100}$ Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different . ( L . D . Kurlyandchik , Leningrad)

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]

Let $n\in \mathbb{N}$ be a number multiple of 4. We take all permutations $(a_1,a_2...a_n)$ of the numbers $(1,2...n)$, for which $\forall j$, $a_i+j=n+1$ where $i=a_j$. Prove that there exist $\frac{(\frac{1}{2}n)!}{(\frac{1}{4}n)!}$ such permutations.

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]

Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that $$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$ where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$

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 $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

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.

Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$. [i]Will Steinberg, United Kingdom[/i]

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

$\indent$ 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 $\frac{n\sqrt n}{1000}$?

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 $p_n(k)$ 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]

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions: \[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}. \]

If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$, $k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$

Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$, where $f$ is a permutation of $\{1, 2, . . . , 6\}$. Let \[\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \} \] Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$.

Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that $$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$ we call $\alpha$ a friendly permutation of $\beta$. Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.

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.

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 $\{0,1,..., n\}$.

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.

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer