Every brick has $5$ holes in a line. The holes can be filled with bolts (fitting in one hole) and braces (fitting into two neighboring holes). No hole may remain free. One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent. How many different such patterns can be produced with $n$ bricks?

Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$

Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.

Let $ n$ be a nonzero positive integer. A set of persons is called a $ n$-balanced set if in any subset of $ 3$ persons there exists at least two which know each other and in each subset of $ n$ persons there are two which don't know each other. Prove that a $ n$-balanced set has at most $ (n \minus{} 1)(n \plus{} 2)/2$ persons.

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers. Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that: [b](1)[/b] $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$. [b](2)[/b] The number of all pairs of integers $(i,j)$ satisfying $1\leq i<j\leq m$ and $a_i>a_j$ is even. For example, when $n=4$, the number of all such lists $(a_1,a_2,\cdots,a_m)$ is $6$, and these lists are $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$.

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

Denote by $S_n$ the group of permutations of the sequence $(1,2,\dots,n).$ Suppose that $G$ is a subgroup of $S_n,$ such that for every $\pi\in G\setminus\{e\}$ there exists a unique $k\in \{1,2,\dots,n\}$ for which $\pi(k)=k.$ (Here $e$ is the unit element of the group $S_n.$) Show that this $k$ is the same for all $\pi \in G\setminus \{e\}.$