From the set of numbers 1 to \(2n\), \(n + 1\) numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.
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.
Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.
A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of
a) more than half of the cards;
b) no less than 20 cards?
You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?