SILK-ROAD 2022

Convex quadrilateral $ABCD$ is inscribed in circle $w.$ Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$ (Kungozhin M.)

Distinct positive integers $A$ and $B$ are given $.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$ (Golovanov A.S.)

In an infinite sequence $\{\alpha\}, \{\alpha^2\}, \{\alpha^3\}, \cdots$ there are finitely many distinct values $.$ Show that $\alpha$ is an integer $. (\{x\}$ denotes the fractional part of $x.)$ (Golovanov A.S.)

In a language $,$ an alphabet with $25$ letters is used $;$ words are exactly all sequences of $($ not necessarily different $)$ letters of length $17.$ Two ends of a paper strip are glued so that the strip forms a ring $;$ the strip bears a sequence of $5^{18}$ letters $.$ Say that a word is singular if one can cut a piece bearing exactly that word from the strip $,$ but one cannot cut out two such non-overlapping pieces $.$ It is known that one can cut out $5^{16}$ non-overlapping pieces each containing the same word $.$ Determine the largest possible number of singular words $.$ (Bogdanov I.)