How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
Write in terms of prime factors the numbers 111, 1111, 11111, 111111, 1111111.
Let \(m\) and \(n\) be integers. Prove that \(mn(m + n)\) is an even number.
Write the following rational numbers in the form of decimal fractions: a) \(\frac {1}{7}\); b) \(\frac {2}{7}\); c) \(\frac{1}{14}\); d) \(\frac {1}{17}\).
Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).
Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
The height of the room is 3 meters. When it was being renovated, it turned out that more paint was needed on each wall than on the floor. Can the area of the floor of this room be more than 10 square meters?