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 $a^2+b^2+c^2$?

Is it possible to arrange the numbers $1,2,...,50$ at the vertices and middles of the sides of a regular $25$-gon so that the sum of the three numbers at the ends and in the middle of each side is the same for all sides?

Jason has $20$ red balls and $14$ bags to store them. Prove that there is a bag, which contains at least two balls.

One of the most useful tools for proving mathematical statements is the Pigeonhole principle. Here is one example: suppose that a flock of $10$ pigeons flies into a set of $9$ pigeonholes to roost. Prove that at least one of these $9$ pigeonholes must have at least two pigeons in it.

Show the following: Pigeonhole principle strong form: Let $q_1,q_2,...,q_n$ be positive integers. If $q_1+q_2+...+q_n-n+1$ objects are put into $n$ boxes, then either the $1$st box contains at least $q_1$ objects, or the $2$nd box contains at least $q_2$ objects, . . ., or the $n$th box contains at least $q_n$ objects.
How can you deduce the usual Pigeonhole principle from this statement?