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 \(S\)?
Izzy wrote a correct equality on the board: \(35 + 10 - 41 = 42 + 12 - 50\), and then subtracted 4 from both parts: \(35 + 10 - 45 = 42 + 12 - 54\). She noticed that on the left hand side of the equation all of the numbers are divisible by 5, and on the right hand side by 6. Then she took 5 outside of the brackets on the left hand side and 6 on the right hand side and got \(5(7 + 2 - 9)4 = 6(7 + 2 - 9)\). Having simplified both sides by a common multiplier, Izzy found that \(5 = 6\). Where did she go wrong?
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.
The natural number \(a\) was increased by 1, and its square increased by 1001. What is \(a\)?
In a basket, there are 30 red and green apples. Among any 12 apples there is at least one red one, and among any 20 apples there is at least one green one. How many red apples and how many green apples are there in the basket?
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
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.)