KGS math club

A group of people on Kiseido Go Server Mathematics room.

The meaning of this page is to collect the problems posed there and save hints and solution suggestions.

Adding problems should be quite straightforward with the copy-paste template in the wiki source. Please add $-tags where required (e.g. ). Still, if you don't, somebody else will; all additions are appreciated.

KGS math problems

Date	Author	Problem	Solutions
20.2.2007	StoneTiger	Does any member of the sequence 1, 4, 20, 80, ... generated by x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3) ever have a factor in common with 2007?	sigmundur
21.6.2008	amkach	Consider the two player game that begins with an even length sequence of positive integers. Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed. A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores). Show that Player One has a non-losing strategy, i.e., can always force a tie or a win.	hints solution solution2
30.6.2008	amkach	For n >= 2, consider the n-dimensional hypercube with side length 4 centered at the origin of n-space. Place inside of it 2^n n-dimensional hyperspheres of radius 1, centered at each of the points (+-1, +-1, ..., +-1). These hyperspheres are tangent to the hypercube and to each other. Then place an n-dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the 2^n hyperspheres of radius 1. In which dimensions n is this central hypersphere contained within the hypercube?	solution
1.7.2008	quimey	Assume m and n are integers and can be expressed as sum of 2 squares (i.e, exists a,b,c,d integers with m=a*a+b*b, n=c*c+d*d). Show m*n can be written as sum of 2 squares. And the same but with 4 squares.	solution
6.7.2008	amkach	Prove or disprove: If P(x) is a polynomial (with non-zero degree) of one real variable and a and b satisfy for all integers n > 0 (i.e., , etc.), then a = b	solution