Given a circle $\omega$ and three different points $A, B, C$ on it. Using a compass and a ruler, construct a point $D$ lying on the circle $\omega$ such that a circle can be inscribed in the quadrilateral $ABCD$ (points $A$, $B$, $C$, $D$ must be located on circle $\omega$ in the indicated order).

How many hexagons are in the figure below with vertices on the given vertices? (Note that a hexagon need not be convex, and edges may cross!) [img]https://cdn.artofproblemsolving.com/attachments/1/9/437add8a9225760e7059b8dc2d481d562a7da2.png[/img]

$ABC$ is a triangle with incenter $I$. The foot of the perpendicular from $I$ to $BC$ is $D$, and the foot of the perpendicular from $I$ to $AD$ is $P$. Prove that $\angle BPD = \angle DPC$. [i]Alex Zhu.[/i]

Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.

$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$. The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$. Show that the circumcenter of $CEF$ lies on $K$.

An equilateral triangle with side 101 is placed on a plane so that one of its sides is horizontal and the triangle is above it. It is divided into smaller equilateral triangles with side 1 by segments parallel to its sides. All sides of these smaller triangles are colored red (including the entire border of the large triangle). An equilateral triangle on a plane is called a "mirror" triangle if its sides are parallel to the sides of the original triangle, but it lies below its horizontal side. What is the smallest number of contours of mirror triangles needed to cover all the red segments? (Mirror triangles may overlap and extend beyond the original triangle.) [i]Proposed by Dmitry Shiryayev[/i]

There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given: \\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$) \\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$). \\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}$

Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any automorphism f for G,there are two automorphisms g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that: (a) Every group which the property $\left( P \right)$ is comutative. (b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property. (c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property. (The order of a finite group is the number of elements of that group).

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

Let triangle $ABC$ have $AB = 5, BC = 8,$ and $\angle ABC = 60^\circ.$ A circle $\omega$ tangent to segments $AB$ and $BC$ intersects segment $CA$ at points $X$ and $Y$ such that points $C, Y , X,$ and $A$ lie along $CA$ in this order. If $\omega$ is tangent to $AB$ at point $Z$ and $ZY \parallel BC,$ compute the radius of $\omega.$

Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.

Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have \[ f(f(f(x)) + f(y)) = f(y) - f(x) \]

For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that \[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]