Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. [i]Proposed by Bojan Bašić, Serbia[/i]

Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.

Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$ \[f(f(x)^2 + f(y)) = xf(x) + y.\]

For $x,y\in\mathbb{R}$, solve the system of equations \[ \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases} \]

Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as $\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad \textbf{(B)}\ -f(y)\qquad \textbf{(C)}\ -f(-y)\qquad \textbf{(D)}\ f(-y)\qquad \textbf{(E)}\ f(y)$

Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.

Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?

Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's [b]not[/b] known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least $1000$ points b) Is it possible that Luffy reaches at least $1000000$ points, [b]without[/b] have less than $-42$ points in any moment?

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers. For $m \geq 1$, we define: \[ X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}. \] Show that \[\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.\]

Let $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$. If the maximum element of $S$ is $p_0/q_0$ in reduced form, find $p_0+q_0$.

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

13) A massless rope passes over a frictionless pulley. Particles of mass $M$ and $M + m$ are suspended from the two different ends of the rope. If $m = 0$, the tension $T$ in the pulley rope is $Mg$. If instead the value m increases to infinity, the value of the tension does which of the following? A) stays constant B) decreases, approaching a nonzero constant C) decreases, approaching zero D) increases, approaching a finite constant E) increases to infinity

A triangle has two sides of lengths $1984$ and $2016$. Find the maximum possible area of the triangle. [i]Proposed by Nathan Ramesh

Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?

If $\sqrt{x+2}=2$, then $(x+2)^2$ equals $\text{(A)}\ \sqrt{2} \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 16$

The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

[color=darkred]For any $ m \in \mathbb{N}^*$ solve the ecuation \[ \left\{\left( x \plus{} \frac {1}{m}\right) ^3\right\} \equal{} x^3 \] [/color]

Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$ and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.

Let $n \ge 3$ be a natural number. Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two. [i]Proposed by Walther Janous[/i]

Compute the sum of all positive integers $a\leq 26$ for which there exist integers $b$ and $c$ such that $a+23b+15c-2$ and $2a+5b+14c-8$ are both multiples of $26$.

Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b, L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides. (N. Beluhov)