Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$? $ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies $$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$ for all real $x$. Prove for all real $x$: [i](a)[/i] $f(x)\ge 4$; [i](b)[/i] $f(x)\ge 7.$

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$.

Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions: (i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$, (ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$, (iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Source: 1976 Euclid Part A Problem 3 ----- The minimum value of the function $2x^2+6x+7$ is $\textbf{(A) } 7 \qquad \textbf{(B) } \frac{5}{2} \qquad \textbf{(C) } \frac{9}{4} \qquad \textbf{(D) } -\frac{9}{2} \qquad \textbf{(E) } \frac{5}{4}$

Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (0)=2022$ and $f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)$ for all $x, y \in \mathbb{R}$.

Consider the function $ h$, defined for all positive real numbers, such that: $$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$ for all $x > 0$. Find $h(x)$ and the value of $h(4)$.

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

Find all functions $f: \mathbb N \to \mathbb N$ for which \[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\] holds for all positive integers $n$.

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

Let us consider a function $f:\mathbb{N}\to\mathbb{N}$ for which $f(1)=1$, $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$. Find the number of values at which the maximum value of $f(n)$ is attained for integer $n$ satisfying $0<n<2014$.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.

Let $a>1$ be a positive integer and let $f(n)=n+[a\{n\sqrt{2}\}]$. Show that there exists a positive integer $n$, such that $f(f(n))=f(n)$, but $f(n) \neq n$.

A particle moves along the $x$-axis. It collides elastically head-on with an identical particle initially at rest. Which of the following graphs could illustrate the momentum of each particle as a function of time? [asy] size(400); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,5)); draw((0,1.5)--(5,1.5)); label("$p$",(0,5),N); label("$t$",(5,1.5),E); label("$\mathbf{(A)}$",(2.5,-0.5)); draw((0,1.5)--(2.5,1.5)--(2.5,0.75)--(4,0.75),black+linewidth(2)); draw((0,3.5)--(2.5,3.5)--(2.5,4.25)--(4,4.25),black+linewidth(2)); draw((8,0)--(8,5)); draw((8,1.5)--(13,1.5)); label("$p$",(8,5),N); label("$t$",(13,1.5),E); label("$\mathbf{(B)}$",(10.5,-0.5)); draw((8,1.5)--(10.5,1.5)--(10.5,2.5)--(12,2.5),black+linewidth(2)); draw((8,3.5)--(10.5,3.5)--(10.5,4.5)--(12,4.5),black+linewidth(2)); draw((16,0)--(16,5)); draw((16,1.5)--(21,1.5)); label("$p$",(16,5),N); label("$t$",(21,1.5),E); label("$\mathbf{(C)}$",(18.5,-0.5)); draw((16,1.5)--(18.5,1.5)--(18.5,2.25)--(20,2.25),black+linewidth(2)); draw((16,3.5)--(18.5,3.5)--(18.5,2.75)--(20,2.75),black+linewidth(2)); draw((24,0)--(24,5)); draw((24,1.5)--(29,1.5)); label("$p$",(24,5),N); label("$t$",(29,1.5),E); label("$\mathbf{(D)}$",(26.5,-0.5)); draw((24,1.5)--(26.5,1.5)--(26.75,3.25)--(28,3.25),black+linewidth(2)); draw((24,3.25)--(26.5,3.25)--(26.75,1.5)--(28,1.5),black+linewidth(2)); draw((32,0)--(32,5)); draw((32,1.5)--(37,1.5)); label("$p$",(32,5),N); label("$t$",(37,1.5),E); label("$\mathbf{(E)}$",(34.5,-0.5)); draw((32,1.5)--(34.5,1.5)--(34.5,0.5)--(36,0.5),black+linewidth(2)); draw((32,3.5)--(34.5,3.5)--(34.5,2.75)--(36,2.75),black+linewidth(2)); [/asy]

Let $ f$ be a linear function for which $ f(6)\minus{}f(2)\equal{}12$. What is $ f(12)\minus{}f(2)$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$, \[ f(f(n-1)) = f (n+1) - f(n)\, ?\]

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.