There are a thousand tickets with numbers 000, 001, ..., 999 and a hundred boxes with the numbers 00, 01, ..., 99. A ticket is allowed to be dropped into a box if the number of the box can be obtained from the ticket number by erasing one of the digits. Is it possible to arrange all of the tickets into 50 boxes?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
When cleaning her children’s room, a mother found \(9\) socks. In a group of any \(4\) of the socks at least two belonged to the same child. In a group of any \(5\) of the socks no more than \(3\) had the same owner. How many children are there in the room and how many socks belong to each child?
Let \(x_1, x_2, \dots , x_n\) be some numbers belonging to the interval \([0, 1]\). Prove that on this segment there is a number \(x\) such that \[\frac{1}{n} (|x - x_1| + |x - x_2| + \dots + |x - x_n|) = 1/2.\]
You are given 8 different natural numbers that are no greater than 15. Prove that there are three pairs of these numbers whose positive difference is the same.
Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.
a) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen such that each ‘L’ shaped group of 3 squares has at least one unshaded square.
b) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen, such that each ‘L’ shaped group of 3 squares has at least one shaded square.
10 school students took part in a Mathematical Olympiad and solved 35 problems in total. It is known that there were students who solved exactly one problem, students who solved exactly two problems, and students who solved exactly three problems. Prove that there is a student who solved exactly 5 problems.