Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$

1) Is it possible to place five circles on the plane in such way that each circle has exactly 5 common points with other circles? 2) Is it possible to place five circles on the plane in such way that each circle has exactly 6 common points with other circles? 3) Is it possible to place five circles on the plane in such way that each circle has exactly 7 common points with other circles?

Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What maximal number of the cells can remain unpainted when the game stops?

Let $H$ be the point of intersection of the altitudes $AD$ and $BE$ of acute triangle $ABC.$ The circles with diameters $AE$ and $BD$ touch at point $L$. Prove that $HL$ is the angle bisector of angle $\angle AHB.$

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$

1) Let $a,b,c\ge0$ and $ab+bc+ca=2.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}+2(a+b+c)\ge6.\]2) Let $a,b,c\ge0$ and $ab+bc+ca=3.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}\ge\frac{3}{2}.\]

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?