In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.

Click for solution We will use with unoriented segments. According to Menelaus' theorem, applied in triangle $ PED$ for the transversal $ \overline{AMC}$, \[ \frac {PA}{AE} \cdot \frac {EC}{CD} \cdot \frac {DM}{MP} = 1, \] and since the quadruple $ (E, B, D, C)$ is harmonic, $ EC/CD = EB/BD$. Therefore, \[ \frac {PA}{AE} \cdot \frac {EB}{BD} \cdot \frac {DM}{MP} = 1, \] and thus, by Ceva's theorem, we conclude that the lines $ EM$, $ DA$, and $ PB$ are concurrent at $ Q$ (i.e. $ PQ$ passes through $ B$).

A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.

The equation $ x^3-ax^2+bx-c=0$ has three (not necessarily different) positive real roots. Find the minimal possible value of $ \frac{1+a+b+c}{3+2a+b}-\frac{c}{b}$.

The sequence $ (x_n)$ is defined as; $ x_1=a$, $ x_2=b$ and for all positive integer $ n$, $ x_{n+2}=2008x_{n+1}-x_n$. Prove that there are some positive integers $ a,b$ such that $ 1+2006x_{n+1}x_n$ is a perfect square for all positive integer $ n$.

$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD=\frac{BD^2}{AB+AD}=\frac{CD^2}{AC+AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD=\frac{DE^2}{CD+CE}$. Prove that $ AE=AB+AC$.

There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists. If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a-good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a-good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n-1)}{n}$.