The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.
Prove that within a group of
Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.
Prove that there is a power of 3 that ends in 001.
A
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.
The numbers
Some whole numbers are placed into a
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.