BG-MO 2015

The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$ , respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$ .

The sequence $a_1, a_2,...$ is defined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$ . Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.

Find all functions $f:\mathbb{R^+}\to\mathbb {R^+}$ such that for all $x,y\in R^+$ the followings hold: $i)$ $f (x+y)\ge f (x)+y$ $ii)$ $f (f (x))\le x$

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$ , respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$ . Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called nice if all students in the group can be ordered in increasing order simultaneously of at least two of the four areas. Find the least positive integer N, such that among any N students there exist a nice group of ten students.