A passenger left his things in an automatic storage room, and when he came to get his things, it turned out that he had forgotten the code. He only remembers that in the code there were the numbers 23 and 37. To open the room, you need to correctly type a five-digit number. What is the least number of codes you need to sort through in order to open the room for sure?
How many nine-digit numbers exist, the sum of the digits of which is even?
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
A sack contains 70 marbles, 20 red, 20 blue, 20 yellow, and the rest black or white. What is the smallest number of marbles that need to be removed from the sack, without looking, in order for there to be no less than 10 marbles of the same colour among the removed marbles.
Some points from a finite set are connected by line segments. Prove that two points can be found which have the same number of line segments connected to them.
There are \(2k+1\) cards numbered with the numbers \(1\) to \(2k+1\). What is the largest number of cards that can be chosen so that no number on a chosen card is equal to the sum of two numbers from two other chosen cards?
We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.
You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.
Will this still always be possible if we are given 1001 integers rather than 1002?
A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.
In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?