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 red, all green and all blue.
This is an extension to a problem my son gave to me. He only asked for rectangles and only used two colours. In fact you can use any finite number of colours.
Just in case, if I’m wrong – prove it.
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) ?
Blame my son for this. I don’t know how to do it (yet).
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 game to be fair?
(Because of the nature of the source, I expect that p = 1/2 isn’t the only answer).
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 next.
(Any resemblance to a real person is entirely intentional).
If a single number, m, is removed from 1, 2, 3, …, n, the average value becomes 40.75.
Find m and n.
This is pretty easy, but requires some thought.
Completely solve n^2 + 20n + 11 = m^2 for integer m, n.
This is very easy if you use the right trick.
Well, maybe a little? The Trick of Mind solution will be to “Name a specific world’s country. A while back, I asked ToM an Easy and True solution about the Colebrook-White with six different equations. A Web site for the county’s computation asked me if they could publish the solution. I told YES, but for FREE. So they did a few weeks ago. Today they told me that about 1000 user have discovered my easy and true solution was right and they thanked me and the country’s web site. The web site told me today 883 folks from the country “love” it!