For real numbers $x$ and $y$, define \[\nabla(x,y)=x-\dfrac1y.\] If \[\underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \dfrac{m}{n}\] for relatively prime positive integers $m$, $n$, compute $100m + n$. [i] Proposed by David Altizio [/i]

Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.

Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does \[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} \] converge?

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) [i]A.Csaszar[/i]

For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation $$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ \{a_i\}_{i \in \mathbb{N}}$ such that $ a_0 \equal{} 1$ e $ a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell$, where $ \ell \equal{} 0$ if $ \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0$, and $ \ell \equal{} 1$ otherwise. How many distinct elements are in the set $ S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}$? [i](Paolo Leonetti)[/i]

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win? [i]Proposed by Matko Ljulj.[/i]

Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties. (i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$. (ii) For every real number $x$ and every integer $n \ge 1$ we have \[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]

Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.

How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?

Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$. a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$. b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves: (i) take a chip from one pile; (ii) take a chip from each of the piles; (ii) remove a chip from one of the piles and put it onto the other. Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.

[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective. [b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.

For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]