The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$, Juku is standing on an escalator step equidistant from each end, facing the direction of travel. He goes by a certain rule: one step forward, two steps back, then again one step forward, two back, etc., taking one every second in increments of one step. Which end will Juku finally get out and at what point will it happen?

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

Find all positive integers N with at most 4 digits such that the number obtained by reversing the order of digits of N is divisible by N and differs from N.

Call a scalene triangle K disguisable if there exists a triangle $K'$ similar to $K$ with two shorter sides precisely as long as the two longer sides of $K$, respectively. Call a disguisable triangle integral if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let $K$ be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of $K$ are perfect squares.

In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.

Father moves $3$ steps forward just as son moves $5$ steps, but this while the father takes $6$ steps, the son does $7$ steps. At first, father and son are together, then the son begins to walk away from his father in a straight line. When the son has done $30$ steps, the father starts to follow him. In a few steps, Dad arrives after the son?

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

Call a k-digit positive integer a hyperprime if all its segments consisting of $ 1, 2, ..., k$ consecutive digits are prime. Find all hyperprimes.

In an exam with k questions, n students are taking part. A student fails the exam if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that (a) all students fail the exam although all questions are easy; (b) no student fails the exam although no question is easy?

Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.