Let $x_0, a, b$ be reals given such that $b > 0$ and $x_0 \ne 0$. For every nonnegative integer $n$ a real value $x_{n+1}$ is chosen that satisfies $$x^2_{n+1}= ax_nx_{n+1} + bx^2_n .$$ a) Find how many different values $x_n$ can take. b) Find the sum of all possible values of $x_n$ with repetitions as a function of $n, x_0, a, b$.

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\] $ \textbf{(A)}\ -2 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.

Given positive integers $n$, $P_1$, $P_2$, …$P_n$ and two sets \[B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}\] A function $f:S \to \mathbb{Z}$ is called [b]Real[/b], if and only if for any positive integers $(y_1,y_2,…,y_n)$ and positive integer $a$ which satisfied $ 1 \leq y_i \leq P_i-a$ $\forall i \in \mathbb{N}$, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*} Find the minimum of $\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|$, where $f$ is a [b]Real[/b] function. [i]Proposed by tob8y[/i]

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent. [i]Author: Arnoldo Aguilar, El Salvador[/i]

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices. What is the smallest number of the edges in the initial graph with 30 vertices?

Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)+1)=y+f(f(x)f(y))$$ for all $x, y \in \mathbb{R}$. (Theresia Eisenkölbl)

Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

In cube $ABCD-A_1B_1C_1D_1$, draw a plane $\alpha$ perpendicular to line $AC'$, and $\alpha$ has intersections with any surface of the cube. The area of the cross section is $S$, the perimeter of the cross section is $l$, then $\text{(A)}$ The value of $S$ is fixed, but the value of $l$ is not fixed. $\text{(B)}$ The value of $S$ is not fixed, but the value of $l$ is fixed. $\text{(C)}$ The value of $S$ is fixed, the value of $l$ is fixed as well. $\text{(D)}$ The value of $S$ is not fixed, the value of $l$ is not fixed either.

Let $A, B,$ and $C$ denote the angles of a triangle. If $\sin^2 A + \sin^2 B + \sin^2 C = 2$, prove that the triangle is right-angled.

All divisors of a positive integer $n$ are listed in the ascending order: $1=d_1<d_2< \ldots < d_k=n$. It turned out that the amount of pairs $(d_i,d_{i+1})$ of adjacent divisors such that $d_{i+1}$ is a multiple of $d_i$, is odd. Prove that $n=pm^2$, where $p$ is the smallest prime divisor of $n$, and $m$ is a positive integer.

Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]

Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$ Determine the [i](a)[/i] maximal; [i](b)[/i] minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

Let $n$ be a positive integer, and let $a>0$ be a real number. Consider the equation: \[ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 \] How many solutions ($x_1, x_2 \cdots , x_n$) does this equation have, such that: \[ 0 \leq x_i \leq a, i \in N^+ \]

If $x$ and $y$ are acute angles such that $x+y=\pi/4$ and $\tan y=1/6$, find the value of $\tan x$. $\textbf{(A) }\dfrac{27\sqrt2-18}{71}\hspace{11.5em}\textbf{(B) }\dfrac{35\sqrt2-6}{71}\hspace{11.2em}\textbf{(C) }\dfrac{35\sqrt3+12}{33}$ $\textbf{(D) }\dfrac{37\sqrt3+24}{33}\hspace{11.5em}\textbf{(E) }1\hspace{15em}\textbf{(F) }\dfrac57$ $\textbf{(G) }\dfrac37\hspace{15.4em}\textbf{(H) }6\hspace{15em}\textbf{(I) }\dfrac16$ $\textbf{(J) }\dfrac12\hspace{15.7em}\textbf{(K) }\dfrac67\hspace{14.8em}\textbf{(L) }\dfrac47$ $\textbf{(M) }\sqrt3\hspace{14.5em}\textbf{(N) }\dfrac{\sqrt3}3\hspace{14em}\textbf{(O) }\dfrac56$ $\textbf{(P) }\dfrac23\hspace{15.4em}\textbf{(Q) }\dfrac1{2007}$

In a certain regular polygon, the measure of each interior angle is twice the measure of each exterior angle. How many sides does this regular polygon have?

(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle (There was an error in published condition of this problem).

For each positive integer $n$, a new number $t_n$ is formed from the numbers $2^n$ and $5^n$ which consists of the digits from $2^n$ followed by the digits from $5^n$. For example, $t_4$ is $16625$. How many digits does the number $t_{2008}$ have?

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

In the diagram below $ \angle CAB, \angle CBD$, and $\angle CDE$ are all right angles with side lengths $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. [asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("$A$",origin,SW); label("$B$",(0,4),dir(160)); label("$C$",(3,0),S); label("$D$",(6,8),dir(80)); label("$E$",(30,-4),E);[/asy]

Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.