In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?
WizardLand Middle School offers a new elective this year: an astrology class. Fifteen students have registered for this course. Prove that at least 2 of these students were born under the same zodiac sign (there are 12 zodiac signs in total, one for each month).
Prove that out of any 11 natural numbers, 2 can be found such that their difference is a multiple of 10.
Eight knights took part in a 3-contest tournament. They competed in archery, sword fighting, and lance throwing. For each contest, a knight was awarded 0, 1 or 2 points. Prove that at least two of these knights earned the same total number of points.
London has more than eight million inhabitants. Show that nine of these people must have the same number of hairs on their heads if it is known that no person has more than one million hairs on his or her head.
Will and Neal are writing numbers on the blackboard. Each number is only composed of digits \(0\) and \(1\) in its decimal representation (as in, normal base 10 numbers). Will then says “I wonder if there is such a number we could write that can be divided by \(2018\)". Is there?
A math circle student Emilio wrote a computer program for his house robot, Basil. Starting from 1, Basil should keep writing bigger and bigger numbers formed by 1s: 1, 11, 111, etc. The program terminates when Basil writes a number that is a multiple of 19. Prove that the program will terminate in fewer than 20 steps.
Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.
16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.
In each cell of a board of size \(5\times5\) a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?