$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.

$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$-gon, and the remaining bulb is placed on the center of said circumference. At any time, the following operations may be applied: [list] [*] Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb. [*] Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.[/list] Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}$$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and $$a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}$$ for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$? $\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

How many distinct real numbers $x$ satisfy the equation $4\cos^3(x)+\sqrt{x}=3\sin(x)+\cos(3x)$?

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

Find all integers $x$ and $y$ such that \[x^3-117y^3=5.\]

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as; $K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$ in the cordinate plane.

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

Points $K$ and $N$ are the midpoints of sides $AC$ and $AB$ of triangle $ABC$. The inscribed circle $\omega$ of the triangle $AKN$ is tangent to $BC$. Find $BC$ if $AC + AB = n$. (Oleksii Karliuchenko)

Let $ABCD$ be a trapezoid such that $AB \parallel CD, \angle{BAC}=25^\circ, \angle{ABC}=125^\circ,$ and $AB+AD=CD.$ Compute $\angle{ADC}.$

Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E$. Let $S$ represent the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC$. If $r=S/S'$, then: $\textbf{(A)}\ r=1\text{ sometimes, }r>1\text{ sometimes} \qquad\\ \textbf{(B)}\ r=1\text{ sometimes, }r<1\text{ sometimes} \qquad\\ \textbf{(C)}\ 0<r<1\qquad \textbf{(D)}\ r>1 \qquad \textbf{(E)}\ r=1 $

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: [i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order; [i](ii)[/i] the polygon is circumscribable about a circle. [i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.