Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.
At the end of the month 5 workers were paid a total of £1,500 between them. Each wants to buy themselves a smartphone that costs £320. Prove that one of them will have to wait another month in order to do so.
Prove that within a group of \(51\) whole numbers there will be two whose difference of squares is divisible by \(100\).
A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.
100 people are sitting around a round table. More than half of them are men. Prove that there are two males sitting opposite one another.
Prove that in any group of 6 people there are either three pairs of people who know one another, or three pairs of people who do not know one another.
A warehouse contains 200 boots of each of the sizes 8, 9, and 10. Amongst these 600 boots, 300 are left boots and 300 are right boots. Prove that there are at least 100 usable pairs of boots in the warehouse.
The alphabet of the Ni-Boom-Boom tribe contains 22 consonants and 11 vowels. A word in this language is defined as any combination of letters in which there are no consecutive consonants and no letter is used more than once. The alphabet is divided into 6 non-empty groups. Prove that it is possible to construct a word from all the letters in the group in at least one of the groups.
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
There are bacteria in a glass. After a second each bacterium divides in half to create two new bacteria. Then after another second these bacteria divide in half, and so on. After a minute the glass is full. After how much time will the glass be half full?