If in a convex quadrilateral $ ABCD, E$ and $ F$ are the midpoints of the sides $ BC$ and $ DA$ respectively. Show that the sum of the areas of the triangles $ EDA$ and $ FBC$ is equal to the area of the quadrangle.

A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $m$ be a positive integer. [b]a.[/b] Show that there exist infinitely many positive integers $k$ such that $1+km^3$ is a perfect cube and $1+kn^3$ is not a perfect cube for all positive integers $n<m$. [b]b.[/b] Let $m=p^r$ where $p \equiv 2 \pmod 3$ is a prime number and $r$ is a positive integer. Find all numbers $k$ satisfying the condition in part a.

Prove that for all positive real numbers $a,b,c$ we have \[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]

Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?

On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.

Determine all possible digits $z$ for which $\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.

Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer $ k$ with the following property: In every coloring satisfying the condition above, it is possible to find a group of $ k$ consecutive points in which all the colors are used.

For every point on the plane, one of $ n$ colors are colored to it such that: $ (1)$ Every color is used infinitely many times. $ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.

There are $50$ male and $50$ female members registered in the QED-DB, who are also there are numbered from $1$ to $100$. In $100$ rounds, Andreas chooses at random one member for the seminar in Bad Tolz, whereupon Katharina already has two each time selected QED members of different sexes may or may not be paired up. Of course QED members cannot be coupled multiple times, ignoring relationships from the time before but both conscientiously. The stability of a relationship between two QED members is the amount of the difference between their numbers in DB and the sum of all stabilities is the promotion of young talent in the QED. What is the greatest possible demand of offspring guaranteed to achieve orgasm? [hide=original wording]In der QED-DB sind 50 m¨annliche und 50 weibliche Mitglieder eingetragen, welche dort mit den Zahlen von 1 bis 100 durchnummeriert sind. In 100 Runden w¨ahlt Andreas jeweils zuf¨allig ein Mitglied fu¨r das Seminar in Bad T¨olz aus, woraufhin jedes Mal Katharina zwei bereits ausgew¨ahlte QEDler unterschiedlichen Geschlechts verkuppeln darf, aber nicht muss. Natu¨rlich k¨onnen QEDler nicht mehrfach verkuppelt werden, Beziehungen aus der Zeit davor ignorieren beide aber gewissenhaft. Die Stabilit¨at einer Beziehung zwischen zwei QEDlern ist der Betrag der Differenz ihrer Zahlen in der DB und die Summe aller Stabilit¨aten ist die Nachwuchsf¨orderung im QED. Was ist die gr¨oßtm¨ogliche Nachwuchsf¨orderung, welche die Orgas garantiert erreichen k¨onnen¿[/hide]

Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have $$2f(ab)=(b+1)f(a)+(a+1)f(b).$$ Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that $$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$ where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime. [i]Authored by Nikola Velov[/i]

Let $s$ and $t$ be two parallel lines. We have marked $k$ points on line $s$ and $n$ points on line $t$ ($k\geq n$). If it is known that the total number of triangles that have their three vertices at marked points is $220$, find all possible values of $k$ and $n$.

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ \quad f(1)=1$ $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$ $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken $f$.

Find the positive integer $ n$ such that \[\arctan\frac{1}{3}\plus{}\arctan\frac{1}{4}\plus{}\arctan\frac{1}{5}\plus{}\arctan\frac{1}{n}\equal{}\frac{\pi}{4}.\]