Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which \[f(f(x)+y)=f(f(y)+x)\] holds for all $x, y\in \mathbb{R}$.

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is: $ \textbf{(A)}\ 36.5 \qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 37.2\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 37.52 $

Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show that $x^3_1$ and $x^3_2$ are the roots of $$y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.$$

Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?

The product of uncommon real roots of the two polynomials $ x^4 \plus{} 2x^3 \minus{} 8x^2 \minus{} 6x \plus{} 15$ and $ x^3 \plus{} 4x^2 \minus{} x \minus{} 10$ is ? $\textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ \minus{} 6 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None}$

Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.

Solve the equation $$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$

[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. [img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img] [b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.) [b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle? [b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$. [b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and \[ x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1. \] Prove that there exists a positive integer $n$ such that $x_n > 1999!$. [i]Ciprian Manolescu[/i]

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $ [i]Ion Tecu[/i]

If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.

For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $? $\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$

$f(x)=x^2+x$ $b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$ Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$

Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.

Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that \[\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. \]

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

Let p>3 be a prime and k>0 an integer. Find the multiplicity of X-1 in the factorization of $ f(X)= X^{p^k-1}+X^{p^k-2}+\cdots+X+1$ modulo p; in other words, find the unique non-negative integer r such that $ (X - 1)^r $ divides f(X) \modulo p, but$ (X - 1)^{r+1} $does not divide f(X) \modulo p.

Find all angles a for which the set of numbers $\sin a$, $\sin 2a$, $\sin 3a$ coincides with the set $cos a$, $cos 2a$, $cos 3a$.

Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers? [i]Proposed by Jacob Weiner[/i]