The sides of a regular polygon of $ n$ sides, $ n > 4$, are extended to form a star. The number of degrees at each point of the star is: $ \textbf{(A)}\ \frac {360}{n} \qquad\textbf{(B)}\ \frac {(n \minus{} 4)180}{n} \qquad\textbf{(C)}\ \frac {(n \minus{} 2)180}{n}$ $ \textbf{(D)}\ 180 \minus{} \frac {90}{n} \qquad\textbf{(E)}\ \frac {180}{n}$

Show that there is a natural number $n$ such that the number $a = n!$ ends exactly in $2009$ zeros.

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

For a point $P$ on the plane, denote by $\lVert P \rVert$ the distance to its nearest lattice point. Prove that there exists a real number $L > 0$ satisfying the following condition: For every $\ell > L$, there exists an equilateral triangle $ABC$ with side-length $\ell$ and $\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}$.

Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that \[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained? $(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$ $(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$ $(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$

Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

There are There are $64$ towns in a country, and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions. but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions.

Let $ABC$ a acute triangle. (a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$: \[\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}\] (b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$); (c) Show that the power of $X$ wrt the circumcircle of $ABC$ is: \[-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4\] Where $a=BC$ , $b=AC$ and $c=AB$.

Consider the sets $A_1,A_2,\dots,A_n$. Set $A_k$ is composed of $k$ disjoint intervals on the real axis ($k=1,2,\dots,n$). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$, and no two of these intervals have a common point.

If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals $ \textbf{(A)}\ -1024 \qquad\textbf{(B)}\ -1024i \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1024 \qquad\textbf{(E)}\ 1024i $

The fraction $ \frac {5x \minus{} 11}{2x^2 \plus{} x \minus{} 6}$ was obtained by adding the two fractions $ \frac {A}{x \plus{} 2}$ and $ \frac {B}{2x \minus{} 3}$. The values of $ A$ and $ B$ must be, respectively: $ \textbf{(A)}\ 5x, \minus{} 11 \qquad\textbf{(B)}\ \minus{} 11,5x \qquad\textbf{(C)}\ \minus{} 1,3 \qquad\textbf{(D)}\ 3, \minus{} 1 \qquad\textbf{(E)}\ 5, \minus{} 11$

A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial