What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ? $ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $

Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that \[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\] for all real numbers $a,b,c,d$. [i]Proposed by usjl[/i]

Let $ a_0$ be a natural number. The sequence $ (a_n)$ is defined by $ a_{n\plus{}1}\equal{}\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$ and $ a_{n\plus{}1}\equal{}[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term.

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. How tall is the building? $\textbf{(A) } 50 \text{ m}\\ \textbf{(B) } 80 \text{ m}\\ \textbf{(C) } 100 \text{ m}\\ \textbf{(D) } 150 \text{ m}\\ \textbf{(E) } 400 \text{ m}$

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$. Prove that $$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}$$ ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$. Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$. Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$.

The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. [i]Proposed by Amirhossein Gorzi[/i]

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

Solve the function $f: \Re \to \Re$: \[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]

Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$: $$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$ Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .

Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that \[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that \[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.

Calculate the $100$th derivative of the function \[\frac{1}{x^2+3x+2}\] at $x=0$ with $10\%$ relative error.

Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.

In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$. [hide="Diagram"][asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(14.4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */ /* draw figures */ draw(circle((-1.32,1.36), 2.98)); draw(circle((3.56,1.53), 3.18)); draw((0.92,3.31)--(-2.72,-1.27)); draw(circle((0.08,0.25), 3.18)); draw((-2.72,-1.27)--(3.13,-0.65)); draw((3.13,-0.65)--(0.92,3.31)); draw((0.92,3.31)--(2.71,-1.54)); draw((-2.41,-1.74)--(0.92,3.31)); draw((0.92,3.31)--(1.05,-0.43)); /* dots and labels */ dot((-1.32,1.36),dotstyle); dot((0.92,3.31),dotstyle); label("$A$", (0.81,3.72), NE * labelscalefactor); label("$c_1$", (-2.81,3.53), NE * labelscalefactor); dot((3.56,1.53),dotstyle); label("$c_2$", (3.43,3.98), NE * labelscalefactor); dot((1.05,-0.43),dotstyle); label("$P$", (0.5,-0.43), NE * labelscalefactor); dot((-2.72,-1.27),dotstyle); label("$B$", (-3.02,-1.57), NE * labelscalefactor); dot((2.71,-1.54),dotstyle); label("$E$", (2.71,-1.86), NE * labelscalefactor); dot((3.13,-0.65),dotstyle); label("$C$", (3.39,-0.9), NE * labelscalefactor); dot((-2.41,-1.74),dotstyle); label("$D$", (-2.78,-2.07), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]

We consider an integer $n \ge 3,$ the set $S=\{1,2,3,\ldots,n\}$ and the set $\mathcal{F}$ of the functions from $S$ to $S.$ We say that $\mathcal{G} \subset \mathcal{F}$ is a generating set for $\mathcal{H} \subset \mathcal{F}$ if any function in $\mathcal{H}$ can be represented as a composition of functions from $\mathcal{G}.$ a) Let the functions $a:S \to S,$ $a(n-1)=n,$ $a(n)=n-1$ and $a(k)=k$ for $k \in S \setminus \{n-1,n\}$ and $b:S \to S,$ $b(n)=1$ and $b(k)=k+1$ for $k \in S \setminus \{n\}.$ Prove that $\{a,b\}$ is a generating set for the set $\mathcal{B}$ of bijective functions of $\mathcal{F}.$ b) Prove that the smallest number of elements that a generating set of $\mathcal{F}$ has is $3.$

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?