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Archive for December, 2013

## Squaring infinity

Posted by Chris under MathsChallenge, Tom (5 Responds)

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. [...]

## p(2014)

Posted by Chris under MathsChallenge (9 Responds)

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) ?

## Little 1

Posted by Chris under MathsChallenge (3 Responds)

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?

## Unfair coin

Posted by Chris under MathsChallenge (14 Responds)

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 [...]

## Party people

Posted by Chris under MathsChallenge (26 Responds)

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.

## Not a prime

Posted by Chris under MathsChallenge (9 Responds)

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

## Slavy’s holiday

Posted by Chris under MathsChallenge (8 Responds)

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 [...]

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