Prove that from any 27 different natural numbers less than 100, two numbers that are not coprime can be chosen.
7 different digits are given. Prove that for any natural number \(n\) there is a pair of these digits, the sum of which ends in the same digit as the number.
In a group of seven boys, everyone has at least three brothers who are in that group. Prove that all seven are brothers.
A council of 2,000 deputies decided to approve a state budget containing 200 items of expenditure. Each deputy prepared his draft budget, which indicated for each item the maximum allowable, in his opinion, amount of expenditure, ensuring that the total amount of expenditure did not exceed the set value of \(S\). For each item, the board approves the largest amount of expenditure that is agreed to be allocated by no fewer than \(k\) deputies. What is the smallest value of \(k\) for which we can ensure that the total amount of approved expenditures does not exceed \(S\)?
A carpet of size 4 m by 4 m has had 15 holes made in it by a moth. Is it always possible to cut out a 1 m \(\times\) 1 m area of carpet that doesn’t contain any holes? The holes are considered to be points.
Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.
Prove that there is a number of the form
a) \(1989 \dots 19890 \dots 0\) (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;
b) \(1988 \dots 1988\), which is divisible by 1989.
A board of size \(2005\times2005\) is divided into square cells with a side length of 1 unit. Some board cells are numbered in some order by numbers 1, 2, ... so that from any non-numbered cell there is a numbered cell within a distance of less than 10. Prove that there can be found two cells with a distance between them of less than 150, which are numbered by numbers that differ by more than 23. (The distance between the cells is the distance between their centres.)
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.
The grandad is twice as strong as the grandma, the grandma is three times stronger than the granddaughter, the granddaughter is four times stronger than the dog, the dog is five times stronger than the cat and the cat is six times stronger than the mouse. The grandad, the grandma, the granddaughter, the dog and the cat together with the mouse can pull out the pumpkin from the ground, which they cannot do without the mouse. How many mice should be summoned so that they can pull out the pumpkin themselves?