a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges? b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges? (Kyiv mathematical festival 2014)

Can an $8\times8$ board be covered with 13 equal 5-celled figures? It's alowed to rotate the figures or turn them over. (Kyiv mathematical festival 2014)

a) There are 8 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D$ such that pairs of teams $A,B$ and $C,D$ won the same number of games in total. b) There are 25 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D,E,F$ such that pairs of teams $A,B,$ $~$ $C,D$ and $E,F$ won the same number of games in total.

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$