$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times. [i](tatari/nightmare)[/i]

Let $S\subset \mathbb{R}$ be a closed set and $f:\mathbb{R}^{2n}\to \mathbb{R}$ be a continuous function. Define a graph $G$ as follows: Let $x$ be a vertex of $G$ iff $x\in \mathbb{R}^{n}$ and $f(x,x)\not\in S$, then connect the vertices $x$ and $y$ by an edge in $G$ iff $f(x,y)\in S$ or $f(y,x)\in S$. Show that the chromatic number of $G$ is countable.

Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ The bisectors of the angles $B$ and $C$ meet each other in $I$ and meet the sides $AC$ and $AB$ in $D$ and $E$, respectively. Prove that $S_{BCDE}=2S_{BIC},$ where $S$ is the area function. [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttqqcc = rgb(0.2,0,0.8); pen qqwuqq = rgb(0,0.39,0); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2); pen ccccff = rgb(0.8,0.8,1); draw((1.89,4.08)--(1.89,4.55)--(1.42,4.55)--(1.42,4.08)--cycle,qqwuqq); draw((1.42,4.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(1.42,4.08),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(4,4.09),fftttt+linewidth(1.2pt)); draw((7.42,4.1)--(1.41,6.24),fftttt+linewidth(1.2pt)); draw((1.41,6.24)--(4,4.09),ccccff+linetype("5pt 5pt")); dot((1.42,4.08),ds); label("$A$", (1.1,3.66),NE*lsf); dot((7.42,4.1),ds); label("$B$", (7.15,3.75),NE*lsf); dot((1.4,10.08),ds); label("$C$", (1.49,10.22),NE*lsf); dot((4,4.09),ds); label("$E$", (3.96,3.46),NE*lsf); dot((1.41,6.24),ds); label("$D$", (0.9,6.17),NE*lsf); dot((3.37,5.54),ds); label("$I$", (3.45,5.69),NE*lsf); clip((-6.47,-7.49)--(-6.47,11.47)--(16.06,11.47)--(16.06,-7.49)--cycle); [/asy]

Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]

Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

The points $P$ and $Q$ lie inside the circle $\omega$. The perpendicular bisector to the segment $PQ$ intersects $\omega$ at points $A$ and $D$. A circle centered on $D$ passing through $P$ and $Q$ intersects $\omega$ at points $B$ and $C$. The segment $PQ$ lies inside the triangle $ABC$. Prove that $\angle ACP = \angle BCQ$. [i]Proposed by A. Zaslavsky [/i]

The positive reals $x_1, x_2, ... , x_n$ have harmonic mean $1$. Find the smallest possible value of $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$.

We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number? [I]Proposed by S. Berlov, S. Ivanov[/i]

Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

Let $\alpha\in(\frac{\pi}{4},\frac{\pi}{2})$, then the order of $(\cos\alpha)^{\cos\alpha},(\sin\alpha)^{\cos\alpha},(\cos\alpha)^{\sin\alpha}$ is $\text{(A)}(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(B)}(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}<(\sin\alpha)^{\cos\alpha}$ $\text{(C)}(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(D)}(\cos\alpha)^{\sin\alpha}<(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}$

I have a $2$ by $4$ grid of squares; how many ways can I shade at least one of the squares such that no two shaded squares share an edge?

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

Let a real-valued sequence $\{x_n\}_{n\geqslant 1}$ be such that $$\lim_{n\to\infty}nx_n=0$$ Find all possible real values of $t$ such that $\lim_{n\to\infty}x_n\big(\log n\big)^t=0$ .