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.
Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.
Cut the interval \([-1, 1]\) into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.
Eight schoolchildren solved \(8\) tasks. It turned out that \(5\) schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.
If each problem is solved by \(4\) pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.
\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).
Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).