It is given parallelogram $ABCD$. On it's sides $AB, BC, CD, DA$ are chosen points $E, F, G, H$ such that area of $EFGH$ is half of the area of $ABCD$. Show that at least one of the quadrilaterals $ABFH$ and $AEGD$ is parallelogram.

Sequences $(a_n), (b_n)$ are defined by $ a_1 = 1, b_1 = 2$, $a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}$, $ b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}$ for all positive integers $n$. Prove that $a_{2020} < 5$.

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

It is given isosceles triangle $ABC$ with $AB = AC$. $AD$ is diameter of circumcircle of triangle $ABC$. On the side $BC$ is chosen point $E$. On the sides $AC, AB$ there are points $F, G$ respectively such that $AFEG$ is parallelogram. Prove that $DE$ is perpendicular to $FG$.

Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure p-horse can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the p-horse is located in one of the unit squares. $a)$ Can the p-horse visit every unit square exactly once? $b$) Can the p-horse visit every unit square exactly once and with the last move return to the initial starting position?