Consider the infinite x,y plane. Colour all the integer coordinate points e.g. (0,0), (1,2) with a red, green or blue dot in any way you please (e.g. randomly). Show that there are an infinite number of squares whose corners are all the same colour.

This is an extension to a problem my son gave to me. [...]

We have a polynomial of order 2012:

p(x) = a2012 x^2012 + a2011 x^2011 + … + a1 x + a0

a2012 etc., are just coefficients.

and p(x) = 1/x for integer x = 1,2,3,…,2013

What is p(2014) ?

As the last two problems might be too hard, here’s a variation of what used to be one of my favourite problems.

What is the smallest natural number that begins with a 1 is tripled when the 1 is moved to the other end?

A coin is biased such that the probability of throwing a H is p, where 0 < p < 1.

Two players, A and B, takes turns to throw the coin. The game ends when either the sequence HHH (then A wins) or HTH (then B wins) is thrown.

What must p be in order for the [...]

In a party of ten people, among any three there are at least two who do not know each other. Show that there are four persons who don’t know each other.

Show that 19*8^n + 17 isn’t a prime, where integer n ≥ 0

Slavy has a 7 day holiday. On any day, he’ll either rest (or recover), drink wine or drink beer. If he never changes drink type on consecutive days, in how many different ways can he enjoy his holiday?

e.g. he could drink wine for 7 days, but he cannot drink wine one day and beer the [...]