For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\] $\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$. Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$. Prove that for any $n\in\mathbb{N}$, every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$.

Three schools $A, B$ and $C$ , each with five players denoted $a_{i}, b_{i}, c_{i}$ respectively, take part in a chess tournament. The tournament is held following the rules: (i) Players from each school have matches in order with respect to indices, and defeated players are eliminated; the first match is between $a_{1}$ and $b_{1}$. (ii) If $y_{j}\in Y$ defeats $x_{i}\in X$ , his next opponent should be from the remaining school if not all of its players are eliminated; otherwise his next oponent is $x_{i+1}$ . The tournament is over when two schools are completely eliminated. (iii) When $x_{i}$ wins a match, its school wins $10^{i-1}$ points. At the end of the tournament, schools $A, B, C$ scored $P_{A}, P_{B}, P_{C}$ respectively. Find the remainder of the number of possible triples $(P_{A}, P_{B}, P_{C})$ upon division by $8.$

Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? [asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i) { for (int j=0; j>-2; --j) { if ((i-j)<3) { add(corner,(50i,50j)); } } } draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy] $\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932$

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$

Given a positive integer $n$, define $f_n(x)$ to be the number of square-free positive integers $k$ such that $kx \le n$. Then, define $v_(n)$ as $$v(n) =\sum^n_{i=1}\sum^n_{j=1}f_n(i^2)- 6f_n (ij) + f_n(j^2).$$ Compute the largest positive integer $2 \le n \le 100$ for which $v(n)-v(n-1)$ is negative. (Note: A square-free positive integer is a positive integer that is not divisible by the square of any prime.)

Find $\lim_{x\to\infty}\left((2x)^{1+\frac1{2x}}-x^{1+\frac1x}-x\right)$.

A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions. [b]Note.[/b] For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$, such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle − a so-called Monge circle − with equation $x^2 + y^2 = a^2 + b^2$.

Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$. For which of these triangles is the product of its altitudes a maximum?

Let $S$ be a finite set. For a positive integer $n$, we say that a function $f\colon S\to S$ is an [i]$n$-th power[/i] if there exists some function $g\colon S\to S$ such that \[ f(x) = \underbrace{g(g(\ldots g(x)\ldots))}_{\mbox{\scriptsize $g$ applied $n$ times}} \] for each $x\in S$. Suppose that a function $f\colon S\to S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x)) = f(x)$ for each $x\in S$?

Let $f:\mathbb N\to (0,\infty)$ satisfy $\prod_{d\mid n} f(d) = 1$ for every $n$ which is not prime. Determine the maximum possible number of $n$ with $1\le n \le 100$ and $f(n)\ne 1$.

Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that [list] [*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal. [*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal. [*]There exist two cells in the grid that do not contain the same number. [/list] Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$. [i]Kiran Reddy[/i]

Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]

For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.

What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?

Determine all $n$ positive integers such that exists an $n\times n$ where we can write $n$ times each of the numbers from $1$ to $n$ (one number in each cell), such that the $n$ sums of numbers in each line leave $n$ distinct remainders in the division by $n$, and the $n$ sums of numbers in each column leave $n$ distinct remainders in the division by $n$.

Find the smallest positive integer $ n$ such that $ 107n$ has the same last two digits as $ n$.

Consider the integer $$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.$$ Find the sum of the digits of $N$.

In trapezoid $ABCD$,$M$ is the midpoint of the base $AD$.Point $E$ lies on the segment $BM$.It is known that $\angle ADB=\angle MAE=\angle BMC$.Prove that the triangle $BCE $ is isosceles.

Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?