Problems

Alice the fox and Basilio the cat have grown $20$ counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has $7$ empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Cut a $7\times 7$ square into $9$ rectangles, out of which you can construct any rectangle whose sidelengths are less than $7$. Show how to construct the rectangles.

There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?

In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.

Find all the prime numbers $p$ such that the number $2p^2+1$ is also prime.

It is known that $a+b+c=5$ and $ab+bc+ac=5$. What are the possible values of $a^2+b^2+c^2$?

Let $r$ be a rational number and $x$ be an irrational number (i.e. not a rational one). Prove that the number $r+x$ is irrational.
If $r$ and $s$ are both irrational, then must $r+s$ be irrational as well?

Definition: We call a number $x$ rational if there exist two integers $p$ and $q$ such that $x=\frac{p}{q}$. We assume that $p$ and $q$ are coprime.
Prove that $\sqrt{2}$ is not rational.

Let $n$ be an integer such that $n^2$ is divisible by $2$. Prove that $n$ is divisible by $2$.

Let $n$ be an integer. Prove that if $n^3$ is divisible by $3$, then $n$ is divisible by $3$.