Given an integer $n\geq 3$, determine the smallest positive number $k$ such that any two points in any $n$-gon (or at its boundary) in the plane can be connected by a polygonal path consisting of $k$ line segments contained in the $n$-gon (including its boundary).

Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.

Given is a convex quadrilateral $ABCD$ with $\angle BAD = \angle BCD$ and $\angle ABC < \angle ADC$. Point $M$ is the midpoint of segment $AC$. Prove that there exist points $X$ and $Y$ on the segments $AB$ and $BC$, respectively, such that $XY \perp BD, MX = MY$ and $\angle XMY = \angle ADC - \angle ABC$. Proposed by Mykhailo Shtandenko

Let $p, q$ and $r$ be positive real numbers such that the equation $$\lfloor pn \rfloor + \lfloor qn \rfloor + \lfloor rn \rfloor = n$$is satisfied for infinitely many positive integers $n{}$. (a) Prove that $p, q$ and $r$ are rational. (b) Determine the number of positive integers $c$ such that there exist positive integers $a$ and $b$, for which the equation $$\left \lfloor \frac{n}{a} \right \rfloor+\left \lfloor \frac{n}{b} \right \rfloor+\left \lfloor \frac{cn}{202} \right \rfloor=n$$is satisfied for infinitely many positive integers $n{}$.

Let $ABC$ be an acute-angled triangle with orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to the line $BC$. Let $T$ be a point on the circle with diameter $AH$ such that this circle is internally tangent to the circumcircle of triangle $BDT$. Let $N$ be the midpoint of segment $AH$. Prove that $BT \perp CN$.

Given is an integer $n \geq 1$ and an $n \times n$ board, whose all cells are initially white. Peter the painter walks around the board and recolors the visited cells according to the following rules. Each walk of Peter starts at the bottom-left corner of the board and continues as follows: - if he is standing on a white cell, he paints it black and moves one cell up (or walks off the board if he is in the top row); - if he is standing on a black cell, he paints it white and moves one cell to the right (or walks off the board if he is in the rightmost column). Peter’s walk ends once he walks off the board. Determine the minimum positive integer $s$ with the following property: after exactly $s$ walks all the cells of the board will become white again.