(a) The sum of some numbers is equal to one. Can it be that the sum of the cubes of these numbers is greater than one?
(b) The sum of some numbers is equal to one. Moreover, it is known that each of the numbers is less than one. Can it be that the sum of the cubes of these numbers is greater than one?
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?
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 six natural numbers, all different, which sum up to 22. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
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?