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 <math>-tags (or dollar signs, it seems) where required, e.g. 
. Still, if you don't, somebody else will; all additions are appreciated.
| Added | 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
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solution |
| 27.7.2008 | royu StoneTiger | You have a collection of 11 balls with the property that if you remove any one of the balls, the other 10 can be split into two groups of 5 so that each weighs the same. If you assume that all of the balls have rational weight, there is a cute proof that they all must weigh the same. Can you find a proof? Can you find a way to extend the result to the general case where the balls have real weights? | solution
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