Problems

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.

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.

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 $n$?

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Does there exist an irreducible tiling with $1\times 2$ rectangles of

(a) $4\times 6$ rectangle;

(b) $6\times 6$ rectangle?

Irreducibly tile a floor with $1\times 2$ tiles in a room that is

(a) $5\times 8$; (b) $6\times 8$.

Having mastered tiling small rooms, Robinson wondered if he could tile big spaces, and possibly very big spaces. He wondered if he could tile the whole plane. He started to study the tiling, which can be continued infinitely in any direction. Can you help him with it?

Tile the whole plane with the following shapes: