[dektep]

เป็นข้อสอบ Team selection test ปี 2008 ของตุรกีครับ
1. 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.

2. 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.

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

4.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 $n$.

5.$D$ 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 $AE = AB + AC$.

6. 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 $k > \frac {m(n - 1)}{n}$.