Do there exist two such triangles that the sides of the first triangle are all less than 1 m, the sides of the second triangle are all greater than 100 m, but the the area of the first triangle is greater than the area of the second triangle?
Does there exist a polygon intersecting each of its own sides only once (each side is intersected only once by a different side) and has all together (a) 6 sides; (b) 7 sides.
Is it possible to make a hole in a wooden cube in such a way that one can drag another cube of the same size through that hole?
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?
In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?
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?
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?
The March Hare and the Dormouse also decided to play a game. They made two piles of matches on the table. The first pile contains 7 matches, and the second one 8. The March Hare set the rules: the players divide a pile into two piles in turns, i.e. the first player divides one of the piles into two, then the second player divides one of the piles on the table into two, then the first player divides one of the piles into two and so on. The loser is the one who cannot not find a pile to divide. The March Hare starts the game. Can the March Hare play in such a way that he always wins?
A board \(7\times7\) is coloured in chessboard fashion in such a way that all the corners are black. The Queen orders the Hatter to colour the board white but sets the rule: in one go it is allowed to repaint only two adjacent cells into opposite colours. The Hatter tries to explain that this is impossible. Can you help the Hatter to present his arguments?