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.

Two play a game on a chessboard $8\times 8$. The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?

Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?

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.

What figure should I put in place of the “?” in the number $888\dots 88?99\dots 999$ (eights and nines are written 50 times each) so that it is divisible by 7?

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

In a certain kingdom there were 32 knights. Some of them were vassals of others (a vassal can have only one suzerain, and the suzerain is always richer than his vassal). A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

(In the kingdom the following law is enacted: “the vassal of my vassal is not my vassal”).

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.