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?
Anna has a garden shaped like an equilateral triangle of side \(8\) metres. She wants to plant \(17\) plants, but they need space – they need to be at least \(2\) metres apart in order for their roots to have access to all the microelements in the ground. Show that Anna’s garden is unfortunately too small.
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.
Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.
The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.
What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size \(8 \times 8\) cells?
Know-it-all came to visit the twin brothers Screw and Nut, knowing that one of them never speaks the truth, and asked one of them: “Are you Screw?”. “Yes,” he replied. When Know-it-all asked the second brother the same question, he received an equally clear answer and immediately determined who was who.
Who was called Screw?
In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be: 1) 15 classmates? 2) 16 classmates?
One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.
In Mexico, environmentalists have succeeded in enacting a law whereby every car should not be driven at least one day a week (the owner informs the police about their car registration number and the day of the week when this car will not be driven). In a certain family, all adults want to travel daily (each for their own business!). How many cars (at least) should the family have, if the family has a) 5 adults? b) 8 adults?