Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.

Find all $n$ such that \[n = d (n) ^ 4\] Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A [i]move[/i] consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.

The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, then $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and $ x \oplus y$ denotes the concatenation of x and y. The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ is defined as above and $ x \oplus y$ is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x. Prove that the two languages consist of the same words.

Let $S$ be a set of positive integers, each of them having exactly $100$ digits in base $10$ representation. An element of $S$ is called [i]atom[/i] if it is not divisible by the sum of any two (not necessarily distinct) elements of $S$. If $S$ contains at most $10$ atoms, at most how many elements can $S$ have?

For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$. [list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$. [*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list] Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps. [i]Proposed by Gerhard Woeginger, Austria[/i]

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. ([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])

Prove that for all positive integers $n,m$, with $m$ odd, the following number is an integer \[ \frac 1{3^mn}\sum^m_{k=0} { 3m \choose 3k } (3n-1)^k. \]

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]