Find all real functions $f$ definited on positive integers and satisying: (a) $f(x+22)=f(x)$, (b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$ for all positive integers $x$ and $y$.

Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that $$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$ for all $n.$ Prove that $$\sum_{n=1}^{\infty} n b_n =a_1.$$

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$. S. Berlov

A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.

The sequence $ < x_n >$ is defined through: $ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$ Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

Consider a triangle $\vartriangle ABC$ with $AB = 7$, $BC = 8$, $CA = 9$, and area $12\sqrt5$. We draw squares on each sides, namely $BCD_2D_1$, $CAE_2E_1$ and $ABF_2F_1$, so that the interiors of the squares do not intersect the interior of the triangle. What is the area of $\vartriangle D_2E_2F_2$?

A two digit prime number is such that the sum of its digits is $13.$ Determine the integer.

Prove that for any function $f(x)$, continuous or otherwise, $$f(f(x)) = x^2 - 1996$$ cannot hold for all real numbers $x$. (S Bogatiy, M Smurov,)

Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$ [asy] draw((0,0)--(8,0)--(4,12)--cycle); draw((8,0)--(1.6,4.8)); label("A", (4,12), N); label("B", (0,0), W); label("C", (8,0), E); label("P", (1.6,4.8), NW); dot((0,0)); dot((4,12)); dot((8,0)); dot((1.6,4.8)); [/asy] $ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} $

Reflect each of the shapes $A,B$ over some lines $l_A,l_B$ respectively and rotate the shape $C$ such that a $4 \times 4$ square is obtained. Identify the lines $l_A,l_B$ and the center of the rotation, and also draw the transformed versions of $A,B$ and $C$ under these operations. [img]https://s8.uupload.ir/files/photo14908574605_i39w.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with the properties: [list=a] [*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $ [*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.

Through point $O$ - the center of a circle circumscribed around an acute triangle - a straight line is drawn, perpendicular to one of its sides and intersecting the other two sides of the triangle (or their extensions) at points $M $ and $N$. Prove that $OM+ON \ge R$, where $R$ is the radius of the circumscribed circle around the triangle.

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

How many ordered triples $(a, b, c)$ of integers $1 \le a, b, c \le 31$ are there such that the remainder of $ab+bc+ca$ divided by $31$ equals $8$?

Given an isosceles triangle $ ABC$ with $ AB \equal{} BC$. A point $ M$ is chosen inside $ ABC$ such that $ \angle AMC \equal{} 2\angle ABC$ . A point $ K$ lies on segment $ AM$ such that $ \angle BKM \equal{}\angle ABC$. Prove that $ BK \equal{} KM\plus{}MC$.

What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? $ \textbf{(A)}\ 23 \qquad\textbf{(B)}\ 26 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ 35 $

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table; - if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$. As a function of $k$, which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$?

Let real angles $\theta_1, \theta_2, \theta_3, \theta_4$ satisfy \begin{align*} \sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4 &= 0, \\ \cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4 &= 0. \end{align*} If the maximum possible value of the sum \[\sum_{i<j}\sqrt{1-\sin\theta_i\sin\theta_j-\cos\theta_i\cos\theta_j}\] for $i, j \in \{1, 2, 3, 4\}$ can be expressed as $a+b\sqrt{c}$, where $c$ is square-free and $a,b,c$ are positive integers, find $a+b+c$ [i]Proposed by Alex Li[/i]

Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.