Problems

Is it possible to cover a $(4n+2)\times (4n+2)$ board with the $L$-tetraminos without overlapping for any $n$? The pieces can be flipped and turned.

Is it possible to cover a $4n\times 4n$ board with the $L$-tetraminos without overlapping for any $n$? The pieces can be flipped and turned.

How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers $54345$ and $12321$ satisfy the condition, but the numbers $23423$ and $56789$ do not.

Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set $C$ consisting of $n$ elements. How many sets can be constructed using the elements of $C$?

A rectangular parallelepiped of the size $m\times n\times k$ is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?

In the Land of Linguists live $m$ people, who have opportunity to speak $n$ languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that $k$ is the maximum number of people, any two of whom can talk without interpreters. It turned out that $11n\le k\le m/2$. Prove that then there are at least $mn$ pairs of people in the country who will not be able to talk without interpreters.

In an $n\times n$ table, two opposite corner squares are black and the rest are white. We wish to turn the whole $n\times n$ table black in two stages. In the first stage, we paint black some of the squares that are white at the moment. In the second stage, we can perform the following two operations as much as we like. The row operation is to swap the colours of all the squares in a particular row. The column operation is to swap the colours of all the squares in a particular column. What is the fewest number of white squares that we can paint in the first stage?

An example of the row operation: let W stand for white and B stand for black and suppose that $n=5$. Also suppose that a particular row has the colours WWBWB. Then performing the row operation would change this row to BBWBW.

How many ways can the numbers $1,1,1,1,1,2,3,\dots ,9$ be listed in such a way that none of the $1$’s are adjacent? The number 1 appears five times and each of $2$ to $9$ appear exactly once.

John’s local grocery store sells 7 kinds of vegetable, 7 kinds of meat, 7 kinds of grains and 7 kinds of cheese. John would like to plan the entire week’s dinners so that exactly one ingredient of each type is used per meal and no ingredients repeat during the week. How many ways can John plan the dinners?

Suppose there is an $7\times 7$ grid. We would like to travel from the bottom left corner to the top right corner in exactly 14 steps. A step is from one point on the grid to another point via a segment of length 1. How many paths are there? The picture below shows one possible path on the grid.