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$.

Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$ Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she erases two numbers $a,b$ (not necessarily different) and write $\gcd(a,b)$. In Bob's round, he erases two numbers $a,b$ (not necessarily different) and write $\mathrm{lcm} (a,b)$. The game ends when only one number remains on the blackboard. Determine the minimal possible $M$ such that Alice could guarantee the remaining number no greater than $M$, regardless of Bob's move.

An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].

A triple $(a,b,c)$ of positive integers is called [b]strong[/b] if the following holds: for each integer $m>1$, the number $a+b+c$ does not divide $a^m+b^m+c^m$. The [b]sum[/b] of a strong triple $(a,b,c)$ is defined as $a+b+c$. Prove that there exists an infinite collection of strong triples, the sums of which are all pairwise coprime.

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.

Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+4)$ is an integer. (Author: O. Podlipski)

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

A natural number $n$ is given. Prove that $$\sum_{n \le p \le n^2} \frac{1}{p} < 2$$ where the sum is across all primes $p$ in the range $[n, n^2]$

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example, to form multidigit numbers (such as $222$) or powers (such as $2^2$). Valid examples: $$\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \;\; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42$$ [i]Proposed by Stephan Wagner, Austria[/i]

Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.

What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None of above} $

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse. [i]Proposed by David Altizio[/i]

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win?

Find all integer solutions of the equation $2x^6 + y^7 = 11$.

Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers.

Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$? A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$

The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.

Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$