a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
In each cell of a \(25 \times 25\) square table, one of the numbers 1, 2, 3, ..., 25 is written. In cells, that are symmetric relative to the main diagonal, equal numbers are written. There are no two equal numbers in any row and in any column. Prove that the numbers on the main diagonal are pairwise distinct.
A coin is tossed three times. How many different sequences of heads and tails can you get?
How many six-digit numbers exist, the numbers of which are either all odd or all even?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?
In a city, there are 15 telephones. Can I connect them with wires so that each phone is connected exactly with five others?
There are 30 people in the class. Can it be that 9 of them have 3 friends (in this class), 11 have 4 friends, and 10 have 5 friends?