Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?
A professional tennis player plays at least one match each day for training purposes. However in order to ensure he does not over-exert himself he plays no more than 12 matches a week. Prove that it is possible to find a group of consecutive days during which the player plays a total of 20 matches.
A country is called a Fiver if, in it, each city is connected by airlines with exactly with five other cities (there are no international flights).
a) Draw a scheme of airlines for a country that is made up of 10 cities.
b) How many airlines are there in a country of 50 cities?
c) Can there be a Fiver country, in which there are exactly 46 airlines?
Can seven phones be connected with wires so that each phone is connected to exactly three others?
Write out in a row the numbers from \(1\) to \(9\) (every number once) so that every two consecutive numbers give a two-digit number that is divisible by \(7\) or by \(13\).
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.
Prove that for any number \(d\), which is not divisible by \(2\) or by \(5\), there is a number whose decimal notation contains only ones and which is divisible by \(d\).
It is known that \[35! = 10333147966386144929 * 66651337523200000000.\] Find the number replaced by an asterisk.