If $z$ is a complex number such that \[ z + z^{-1} = \sqrt{3}, \] what is the value of \[ z^{2010} + z^{-2010} \, ? \]

A graph is called a [i]self-intersecting [/i]graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings. a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting. b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$. c) Find all self-intersecting graphs. [img]http://i35.tinypic.com/x43s5u.png[/img]

Two friends walked towards each other along a straight road. Each had a constant speed but one was faster than the other. At one moment each friend released his dog to run freely forward, the speed of each dog is the same and constant. Each dog reached the other person and then returned to its owner. Which dog returned to its owner the first, of the person who walks fast or who walks slow?

Ana and Bruno have an $8 \times 8$ checkered board. Ana paints each of the $64$ squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the $4$ squares where they intersect. Bruno's goal is for these $4$ squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.

[b]4.[/b] Find all functions of two variables defined over the entire plane that satisfy the relations $f(x+u,y+u)=f(x,y)+u$ and $f(xv,yv)= f(x,y) v$ for any real numbers $x,y,u,v$. [b](R.12)[/b]

Find the natural numbers $ n $ which have the property that $ \log_2 \left( 1+2^n \right) $ is rational. [i]Cornel Berceanu[/i]

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.

Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

(a) Let $p>1$ a real number. Find a real constant $c_p$ for which the following statement holds: If $f: [-1,1]\rightarrow\mathbb{R}$ is a continuously differentiable function with $f(1)>f(-1)$ and $|f'(y)|\le1 \forall y\in[-1,1]$, then $\exists x\in[-1,1]: f'(x)>0$ so that $\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|$. (b) What if $p=1$?

Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions: [b]I.[/b] $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$. [b]II.[/b] Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$. (By $(AB)$ we denote vector $AB$)

$a$, $b$, $c$, $d$ are positive integers and \[ad=b^{2}+bc+c^{2}\] Prove that \[a^{2}+b^{2}+c^{2}+d^{2}\] is a composed number.

For how many positive integers $m$ is \[\dfrac{2002}{m^2-2}\] a positive integer? $\textbf{(A) }\text{one}\qquad\textbf{(B) }\text{two}\qquad\textbf{(C) }\text{three}\qquad\textbf{(D) }\text{four}\qquad\textbf{(E) }\text{more than four}$

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities \begin{align*}4n-18&<2008,\\7n+17&>2008.\end{align*}

[b]3.[/b] A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. [b](C. 4)[/b]

Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old? $\textbf{(A)}\ 134\qquad \textbf{(B)}\ 155 \qquad \textbf{(C)}\ 176\qquad \textbf{(D)}\ 194\qquad \textbf{(E)}\ 243$

How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$? $ \mathrm{(A) \ } 58 \qquad \mathrm{(B) \ } 59 \qquad \mathrm {(C) \ } 60 \qquad \mathrm{(D) \ } 61 \qquad \mathrm{(E) \ } 62$

For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.

Find all real numbers $x$ for which the following equality holds : $$\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}$$

Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Let $\omega_1$ and $\omega_2$ be two circles that intersect at two points: $A$ and $B$. Let $C$ and $E$ be on $\omega_1$, and $D$ and $F$ be on $\omega_2$ such that $CD$ and $EF$ meet at $B$ and the three lines $CE$, $DF$, and $AB$ concur at a point $P$ that is closer to $B$ than $A$. Let $\Omega$ denote the circumcircle of $\triangle DEF$. Now, let the line through $A$ perpendicular to $AB$ hit $EB$ at $G$, $GD$ hit $\Omega$ at $J$, and $DA$ hit $\Omega$ again at $I$. A point $Q$ on $IE$ satisfies that $CQ=JQ$. If $QJ=36$, $EI=21$, and $CI=16$, then the radius of $\Omega$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a, c) = 1$. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]