Is it true that for any point inside any convex quadrilateral the sum of the distances from the point to the vertices of the quadrilateral is less than the perimeter?
A rectangle is made up from six squares. Find side length of the largest square if side length of the smallest square is 1.
There are eight points inside a circle of radius 1. Show that there are at least two points with distance between them less then 1.
This shape below is made up from squares.
Find side length of the bottom square if side length of the smallest square is equal to 1.
In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?
There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
The March Hare bought seven drums of different sizes and seven drum sticks of different sizes for his seven little leverets. If a leveret sees that their drum and their drum sticks are bigger than a sibling’s, they start drumming as loud as they can. What is the largest number of leverets that may be drumming together?
You are given a convex quadrilateral. Is it always possible to cut out a parallelogram out of the quadrilateral such that three vertices of the new parallelogram are the vertices of the old quadrilateral?
A strange wonderland creature is called a painting chameleon. If the queen puts the painting chameleon on a chess-like board then he moves one square at a time along the board either horizontally or vertically. When he moves, he either changes his colour to the colour of the square he moves to, or he paints the square he moves to into his own colour. The queen puts a white painting chameleon on an all-black board \(8\times8\) and orders the chameleon to paint the board into a chessboard. Can he succeed?
Mary Ann and Alice went to buy some cupcakes. There are at least five different types of cupcakes for sale (all different types are priced differently). Mary Ann says, that whatever two cupcakes Alice buys, Mary Ann can always buy another two cupcakes spending the same amount of money as Alice. What should be the smallest number of cupcakes available for sale at the shop if Mary Ann is not lying?