KURSCHAK 1989

In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$ , define $A=\ell\cap a$ , and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$ . Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.

For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$ .

We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$ , in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$ . There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).

Prove that starting from the point $\left(1;\sqrt 2\right)$ , we cannot return to it in finitely many steps.