Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.
2006 Turkey MO (2nd round)
Day 1
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where \[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]
Day 2
$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$
$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly. Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.