Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?
The numbers \(1, 2, 3, \dots , 99\) are written onto 99 blank cards in order. The cards are then shuffled and then spread in a row face down. The numbers \(1, 2, 3, \dots, 99\) are once more written onto in the blank side of the cards in order. For each card the numbers written on it are then added together. The 99 resulting summations are then multiplied together. Prove that the result will be an even number.
The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.
The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.
The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.
Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Prove that amongst the numbers of the form \[19991999\dots 19990\dots 0\] – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.
Do you think that among the four consecutive natural numbers there will be at least one that is divisible a) by 2? b) by 3? c) by 4? d) by 5?
Six sacks of gold coins were found on a sunken ship of the fourteenth century. In the first four bags, there were 60, 30, 20 and 15 gold coins. When the coins were counted in the remaining two bags, someone noticed that the number of coins in the bags has a certain sequence. Having taken this into consideration, could you say how many coins are in the fifth and sixth bags?