Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

Compute the sum $\sum^{200}_{n=1}\frac{1}{n(n+1)(n+2)}$ .

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.

Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>b$ and no square dividing $a$ or $b$. Find $100a+10b+c$. [i]Proposed by Michael Kural[/i]

A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers. $a)$ Does there exist a meaningful set which consists of $2019$ numbers? $b)$ Does there exist an infinite meaningful set? Note: The geometric mean of the non-negative numbers $a_1, a_2,\cdots, $ $a_n$ is defined as $\sqrt[n]{a_1a_2\cdots a_n} .$

Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality: $$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$ (Karl Czakler)

p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$. p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ? p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements. p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of ​​the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$. p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.

Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$? [i]J. Kalina, L. Lempert[/i]

Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.

Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$ may achieve and find all $a,b,c$ for which equality occurs.\\ \\ [i](Andrei Bâra)[/i]

At 12 o'clock in the afternoon, "Zaporozhets" and "Moskvich" were at a distance of 90 km and began to move towards each other at a constant speed. Two hours later they were again at a distance of 90 km. Dunno claims that ''Zaporozhets'' before meeting with ''Moskvich'' and ''Moskvich'' after the meeting with ''Zaporozhets'' , have drove a total of 60 km. Prove that he is wrong. [hide=original wording]В 12 часов дня ''Запорожец'' и ''Москвич'' находилисьна расстоянии 90 км и начали двигаться навстречу друг другу с постоянной скоростью. Через два часа они снова оказались на расстоянии 90 км. Незнайка утверждает, что ''Запорожец'' до встречи с ''Москвичом'' и ''Москвич'' после встречи с ''Запорожцем'' проехали в сумме 60 км. Докажите, что он не прав. [/hide]

Given three non-negative real numbers $a$, $b$ and $c$. The sum of the modules of their pairwise differences is equal to $1$, i.e. $|a- b| + |b -c| + |c -a| = 1$. What can the sum $a + b + c$ be equal to?

Study the convergence of a sequence defined by $u_0\ge0$ and $u_{n+1}=\sqrt{u_n}+\frac1{n+1}$ for all $n\in\mathbb N_0$.

A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is $40$. Which values ​​are possible for the sum of the red numbers? (G. Kirchner, University of Innsbruck)

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for: $\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$ $\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Prove that $\sqrt 2 +\sqrt 3 +\sqrt{1990}$ is irrational.

Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by $$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$ and more generally \[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \] Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$? $ \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 $

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]

Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$.