Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$ \[x_n=2x_{n-1}-4x_{n-2}+3.\] Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

Let $p$ be an odd prime. Suppose that positive integers $a,b,m,r$ satisfy $p\nmid ab$ and $ab > m^2$. Prove that there exists at most one pair of coprime positive integers $(x,y)$ such that $ax^2+by^2=mp^r$.

Ana and Celia sell various objects and obtain for each object as many euros as objects they sold. The money obtained is made up of some $10$ euro bills and less than $10$ coins of $1$ euro . They decide to distribute the money as follows: Ana takes a $10$ euro bill and then Celia, and so on successively until Ana takes the last $10$ euro note, and Celia takes all the $1$ euro coins . How many euros more than Celia did Ana take? Give all the possibilities. [hide=original wording]Ana y Celia venden varios objetos y obtienen por cada objeto tantos euros como objetos vendieron. El dinero obtenido está constituido por algunos billetes de 10 euros y menos de 10 monedas de 1 euro. Deciden repartir el dinero del siguiente modo: Ana toma un billete de 10 euros y después Celia, y así sucesivamente hasta que Ana toma el último billete de 10 euros, y Celia se lleva todas las monedas de 1 euro. ¿Cuántos euros más que Celia se llevó Ana? Dar todas las posibilidades.[/hide]

Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$

To every natural number $k, k \geq 2$, there corresponds a sequence $a_n(k)$ according to the following rule: \[a_0 = k, \qquad a_n = \tau(a_{n-1}) \quad \forall n \geq 1,\] in which $\tau(a)$ is the number of different divisors of $a$. Find all $k$ for which the sequence $a_n(k)$ does not contain the square of an integer.

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1$ is perfect square iff $ m \equal{} n$.

What are the last two digits of $$2^{3^{4^{...^{2019}}}} ?$$

Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

Solve the equation for positive integers $m, n$: \[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions: (1) $x_0=1, x_l=x$; (2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ . Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.

Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$