The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.
2007 Bulgaria National Olympiad
Day 1
Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$. A. Ivanov, E. Kolev
Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$. O. Mushkarov, N. Nikolov Click to reveal hidden textNo-one in the competition scored more than 2 points
Day 2
Let $k>1$ be a given positive integer. A set $S$ of positive integers is called good if we can colour the set of positive integers in $k$ colours such that each integer of $S$ cannot be represented as sum of two positive integers of the same colour. Find the greatest $t$ such that the set $S=\{a+1,a+2,\ldots ,a+t\}$ is good for all positive integers $a$. A. Ivanov, E. Kolev
Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1. O. Mushkarov, N. Nikolov
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.