Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$

For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?

Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that $|6n^2-17n-39|=p^k$

A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents. This is represented by the figure below. [asy] import olympiad; pair A, B, C; A = dir(90); B = dir(210); C = dir(330); draw(A--B--C--cycle); draw(incircle(A,B,C)); draw(incircle(A,2/3*A+1/3*B,2/3*A+1/3*C)); draw(incircle(A,8/9*A+1/9*B,8/9*A+1/9*C)); draw(incircle(A,26/27*A+1/27*B,26/27*A+1/27*C)); draw(incircle(A,80/81*A+1/81*B,80/81*A+1/81*C)); draw(incircle(A,242/243*A+1/243*B,242/243*A+1/243*C)); draw(incircle(B,2/3*B+1/3*A,2/3*B+1/3*C)); draw(incircle(B,8/9*B+1/9*A,8/9*B+1/9*C)); draw(incircle(B,26/27*B+1/27*A,26/27*B+1/27*C)); draw(incircle(B,80/81*B+1/81*A,80/81*B+1/81*C)); draw(incircle(B,242/243*B+1/243*A,242/243*B+1/243*C)); draw(incircle(C,2/3*C+1/3*B,2/3*C+1/3*A)); draw(incircle(C,8/9*C+1/9*B,8/9*C+1/9*A)); draw(incircle(C,26/27*C+1/27*B,26/27*C+1/27*A)); draw(incircle(C,80/81*C+1/81*B,80/81*C+1/81*A)); draw(incircle(C,242/243*C+1/243*B,242/243*C+1/243*A)); [/asy] What is the ratio of the area of the largest circle to the combined area of all the other circles? $\text{(A) }\frac{8}{1}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }\frac{9}{1}\qquad\text{(D) }\frac{9}{3}\qquad\text{(E) }\frac{10}{3}$

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Prove that in all triangles $\vartriangle ABC$ with $\angle A = 2 \angle B$ it holds that, if $D$ is the foot of the perpendicular from $C$ to the perpendicular bisector of $AB$, $\frac{AC}{DC}$ is constant for any value of $\angle B$.

Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $ [b]a)[/b] Verify that $ G $ is a group. [b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $

Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties: [list] [*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and [*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$. [/list] [i]Viktor Simjanoski, Macedonia[/i]

Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

The triangle $ ABC$ with the circumcircle $ k(U,r)$ is given. On the extension of the radii $ UA$ a point $ P$ is chosen. The reflection of the line $ PB$ on the line $ BA$ is called $ g$. Likewise the reflection of the line $ PC$ on the line $ CA$ is called $ h$. The intersection of $ g$ and $ h$ is called $ Q$. Find the geometric location of all possible intersections $ Q$, while $ P$ passes through the extension of the radii $ UA$.

Prove for every positive integer $n{:}$ \[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

In a city of gnomes there are $1000$ identical towers, each of which has $1000$ stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of $1000$ possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the $k$-th story of his tower and the other one in the $(k+1)$-st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends. [i]Authored by Nikola Velov[/i]