Let $a>0$. Show that $$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$ lies between $e^a$ and $e^{a+1}.$

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let ${\cal L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any points $p,q$ of $S$, the number of lines in ${\cal L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour. Note: A line $\ell$ separates two points $p$ and $q$ if $p$ and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.

Find all pairs of positive real numbers $(a,b)$ such that $\frac{n-2}{a} \leq \left\lfloor bn \right\rfloor < \frac{n-1}{a}$ for all positive integes $n$.

Let $x, y, z$ be positive real numbers satisfy the condition $x^2 +y^2 + z^2 = 2(x y + yz + z x)$. Prove that $x + y + z + \frac{1}{2x yz} \ge 4$

There are $5$ boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations: i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box, ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box. Is it possible, that after a few steps, obtained conditions where each box contains exactly $17^{5^{2016}}$ balls?

Fix an integer $n \geq 3$. Let $\mathcal{S}$ be a set of $n$ points in the plane, no three of which are collinear. Given different points $A,B,C$ in $\mathcal{S}$, the triangle $ABC$ is [i]nice[/i] for $AB$ if $[ABC] \leq [ABX]$ for all $X$ in $\mathcal{S}$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several nice triangles). A triangle is [i] beautiful [/i] if its vertices are all in $\mathcal{S}$ and is nice for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ beautiful triangles.

How many integer pairs $(m,n)$ are there satisfying $4mn(m+n-1)=(m^2+1)(n^2+1)$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$

Let $a, b$ and $c$ be integers such that $a+b+c = 0$. Prove that $\frac12(a^4 +b^4 +c^4)$ is a perfect square.

A bird flying in a straight line, initially at $\text{10 m/s}$, uniformly increases its speed to $\text{18 m/s}$ while covering a distance of $\text{40 m}$. What is the magnitude of the acceleration of the bird? (a) $\text{0.1 m/s}^2$ (b) $\text{0.2 m/s}^2$ (c) $\text{2.0 m/s}^2$ (d) $\text{2.8 m/s}^2$ (e) $\text{5.6 m/s}^2$

Determine the remainder when $$2^{\frac {1 \cdot 2}{2}} + 2^{\frac {2 \cdot 3}{2}}+ ...+ 2^{\frac {2011 \cdot 2012}{2}}$$ is divided by $7$.

The differentiable function $F:\mathbb{R}\to\mathbb{R}$ satisfies $F(0)=-1$ and \[\dfrac{d}{dx}F(x)=\sin (\sin (\sin (\sin(x))))\cdot \cos( \sin (\sin (x))) \cdot \cos (\sin(x))\cdot\cos(x).\] Find $F(x)$ as a function of $x$.

Let $ABC$ be an isosceles triangle ($AB=AC$). The altitude $AH$ and the perpendiculare bisector $(e)$ of side $AB$ intersect at point $M$ . The perpendicular on line $(e)$ passing through $M$ intersects $BC$ at point $D$. If the circumscribed circle of the triangle $BMD$ intersects line $(e)$ at point $S$ , the prove that: a) $BS // AM$ . b) quadrilateral $AMBS$ is rhombus.

* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.

Fix positive integers $r>s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r-1$ that shares at least $s$ elements with each set in $F$.

P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?

Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square. ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$)

$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that \[ \left\lvert x\right\rvert + \left\lvert y + \frac{1}{2} \right\rvert < n. \] A path is a sequence of distinct points $(x_1 , y_1), (x_2, y_2), \ldots, (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \ldots, \ell$, the distance between $(x_i , y_i)$ and $(x_{i-1} , y_{i-1} )$ is $1$ (in other words, the points $(x_i, y_i)$ and $(x_{i-1} , y_{i-1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).

There is a square $ ABCD. $ $ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $ They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $ Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $ If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.