2The Pigeonhole Principle: If (k+1) or more objects are placed into k boxes, then there isat least one box containing two or more objects.holesobjectsTo prove this statement, let us suppose that each box containsless than 2 objects; then the total number of objects would beless than k, contradiction.
3Problem 1 15 tourists tried to hike the Washington mountain. The oldestof them is 33, whilethe youngest one is20. Prove that thereare 2 tourists of thesame age.
4Problem 2 Wizard told Dorothy that he would help her to get home, if she could create a magic 6x6square with entries either “+1”or “-1”, so that all vertical,horizontal and diagonal sumswould be different.Prove that the Wizard cannothelp Dorothy, because there isno such square.-11
5Problem 3 The ocean covers more than half of the Earth surface. Can youprove that there are twopoints in the ocean whichare located on the exactlyopposite ends of an Earthdiameter?
6Problem 4 There are 30 students in the classroom. Peter got the worth results on atest, and he made 13mistakes. Can you provethat there are at least 3students who all made thesame number of mistakes(not necessarily 13)?
7Problem 5 John has 30 socks in a box: 10 white, 10 red and 10 black. How many socksmust he pull out withoutlooking, in order to beguaranteed to have:1)two socks of the same color2) two black socks3) two different socks
8Problem 6 There are 380 students at Magic school. Prove that there are atleast two students whosebirthdays happen on asame day.
9Problem 7 There are 4,000,000,000 humans in our world which are less than100 years old. Prove thatthere are at least twopersons that have beenborn at the same second.
10Problem 8 A student drew 12 not parallel lines on the sheet of paper. Prove that there are atleast two lines thatmake an angle of lessthan 15 degrees withone another.
11Problem 9 A student has chosen 52 natural numbers. Prove that you can choosetwo from the list, so thateither their sum or theirdifference would bedivisible by 100.
12Problem 10 65 students wrote three tests. The possible grades are: A, B, C and D.Prove that there areat least two of themwho managed to getthe same grades for allthree tests.