Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$

There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule: In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy. [i](Proposed by Theresia Eisenkölbl)[/i]

Let $f(x_{1}, x_{2}, . . . , x_{n})$ be a polynomial with integer coefficients of degree less than $n$. Prove that if $N$ is the number of $n$-tuples $(x_{1}, . . . , x_{n})$ with $0 \leq x_{i} < 13$ and $f(x_{1}, . . . , x_{n}) = 0 (mod 13)$, then $N$ is divisible by 13.

Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

Consider the sequence $x_n>0$ defined with the following recurrence relation: \[x_1 = 0\] and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\] Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.

Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.

Let $m,n\ge 2$ be given integers. Prove that there exist positive integers $a_1<a_2<\ldots<a_m$ so that for any $1\le i<j\le m$ the number $\frac{a_j}{a_j-a_i}$ is an integer divisible by $n$.

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

$ a \minus{} )$ Find all prime $ p$ such that $ \dfrac{7^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square $ b \minus{} )$ Find all prime $ p$ such that $ \dfrac{11^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square

We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$, and $ \Delta(s) <\Delta(b)$. a) Prove that the following mapping is a degree mapping: \[ \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n\] b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$. c) Prove that $ \delta \equal{}\Delta_{0}$ [img]http://i16.tinypic.com/4qntmd0.png[/img]

Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions: $(i)$ $a_0=a_n=1$; $(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$; $(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$; $(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$. Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.

For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

Compute the sum of all positive integers $n$ with the property that $x^n \equiv 1$ (mod $2016$) has $n$ solutions in $\{0, 1, 2, ... , 2015\}$.

You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$? Here $(x, y)$ denotes the greatest common divisor of $x, y$. [i]Proposed by Anton Trygub[/i]

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

Show that for every integer $n \ge 3$ there exists positive integers $x_1, x_2, . . . , x_n$, pairwise different, so that $\{2, n\} \subset \{x_1, x_2, . . . , x_n\}$ and $$\frac{1}{x_1}+\frac{1}{x_2}+.. +\frac{1}{x_n}= 1.$$

Find the largest $ 85$-digit integer which has property: the sum of its digits equals to the product of its digits.

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \$1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}\] Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.