Problems

The function $F$ is given on the whole real axis, and for each $x$ the equality holds: $F(x+1)F(x)+F(x+1)+1=0$.

Prove that the function $F$ can not be continuous.

a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the $k$-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...

a) Where in this sequence will the 1000th letter “A” be?

b) Prove that this sequence is non-periodic.

Solve the equation $x+\frac{1}{(y+1/z)}=10/7$ in natural numbers.

A numerical sequence is defined by the following conditions: $$a_1=1,a_{n+1}=a_n+\lfloor \sqrt{a_n}\rfloor .$$

Prove that among the terms of this sequence there are an infinite number of complete squares.

The function $f(x)$ on the interval $[a,b]$ is equal to the maximum of several functions of the form $y=C\times {10}^{-|x-d|}$ (where $d$ and $C$ are different, and all $C$ are positive). It is given that $f(a)=f(b)$. Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.

Let $n$ numbers are given together with their product $p$. The difference between $p$ and each of these numbers is an odd number.

Prove that all $n$ numbers are irrational.

a) Could an additional $6$ digits be added to any $6$-digit number starting with a $5$, so that the $12$-digit number obtained is a complete square?

b) The same question but for a number starting with a $1$.

c) Find for each $n$ the smallest $k=k(n)$ such that to each $n$-digit number you can assign $k$ more digits so that the resulting $(n+k)$-digit number is a complete square.

Are there such irrational numbers $a$ and $b$ so that $a>1$, $b>1$, and $\lfloor a^m\rfloor$ is different from $\lfloor b^n\rfloor$ for any natural numbers $m$ and $n$?

Two players in turn paint the sides of an $n$-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what $n$ can the second player win, no matter how the first player plays?