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. In order to write something, I'm afraid you need to register to the AoPS wiki first. After that you're good to go.

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. $f''(x)$ . Still, if you don't, somebody else will; all additions are appreciated.

KGS math problems
Added	Author	Problem	Solutions
2011-11-08	Niall	An island has 3 colours of snake: red, blue and green. When snakes of different colors meet they both turn into the third color. They never breed or die. We start with 13 red, 15 green, and 17 blue snakes. Show how to achieve a state where all the snakes are the same color, or prove it is impossible.
2011-08-30	maproom	You have three amply large buckets, each containing a known number of pebbles. You are allowed, as often as you like, to choose two buckets and to move from the first to the second as many pebbles as were previously in the second. You must always choose them so that there are enough pebbles in the first. Show that, for all sets of starting numbers, you can eventually obtain an empty bucket.
2011-08-30	parigi	Arrange a bridge pack in a 13×4 array such that (i) each row has one of each rank, (ii) each row has three or four of each suit, (iii) each column has one of each suit, and (iv) each pair of distinct ranks appears together in some column.	Unsolved
2011-07-07	maproom	N dwarfs, who can discuss strategy first, each have an ordered infinity of red and blue hats placed on their heads, colours assigned randomly. They can see each others', but not their own, hats. Each is to specify a hat on their head (e.g. hat number 4) with a single simultaneous guess. Success is group-wise: they succeed iff everyone manages to identify a blue hat on their own head. How well can they do?	solution
2011-05-??	maproom	Label each of two 6-sided dice with a distinct positive integer on each face, so that all 36 sums that can be obtained from a throw of the dice are prime. Choose the 12 numbers so as to minimise their sum. Label each of two 6-sided dice with a distinct positive integer on each face, so that all 36 sums that can be obtained from a throw of the dice are prime. Choose the 12 numbers so as to minimise their sum.	solution by jj
2011-04-24	maproom	The "derived graph" of a given graph is defined as a graph with the same vertices, and an edge joining any two vertices that are two edges but not less apart in the original graph. What are the derived graphs of (i) the pentagon? (ii) the cube? (iii) the icosahedron?	solution
2011-03-20	parigi	A large circular track has only one place where horsemen can pass; many can pass at once there. Is it possible for nine horsemen to gallop around it continuously, all at different constant speeds?	hint, solution, explicit solution by iceweasel
2011-03-06	iceweasel	A 52-card deck is shuffled and cards are taken from the top and shown, one by one. You are forced to make a $1 bet that "the next card drawn will be black" once before the deck is emptied. Your only freedom is choosing when to make this bet, depending on what you've seen so far. What is the maximum expected gain from your bet?	solution
2011-02-14	warfreak2	A regular tetrahedron formed from six thin sticks is completely infested with greenfly, which breed rapidly and spread along the sticks at 1mm per second. There are three ladybugs that can walk at up to 1.1 mm per second, eating the greenfly that they pass. How can they exterminate the greenfly?	solution by Swifft
Feb. 2011	maproom	The number of ways to choose k things from n (n>2k) is equal to the number of ways to choose n-k things. Find a general way to pair up the k-member subsets with the (n-k)-member subsets such that each of the former is a subset of its partner.	solution by iceweasel
Jan. 2011	maproom	How many dissimilar ways are there to arrange five points in the plane such that there are only two distinct distances between pairs of the points?	solution by Warfreak2
late 2010		Find a set P such that P×P is a proper subgroup of P.	solution by maproom
11.8.2010	ghej	For the curve x^2 + x y + y^2 = 1, find the tangent that passes through the point (0,2).	solution
19.8.2009	royu	You have the set {9, 99, 999, ...}. Show that given any natural number n not divisible by 2 or 5, n divides at least one element of the set.	hint solution
19.8.2009	bourbaki	Suppose A and B are n x n matrices with real entries such that either A or B commutes with C = AB - BA. Prove that C is nilpotent, i.e. C^k = 0 for some integer k	solution
18.7.2009	taoyan	How many times do the clock hands (hour and minute) overlap between 11:59:59 before lunch and 00:00:01 at night?	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
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 $P^{(n)}(a) = P^{(n)}(b)$ for all integers n > 0 (i.e., $P(a) = P(b), P'(a) = P'(b), P''(a) = P''(b)$ , etc.), then a = b	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 $m=a^2+b^2, n=c^2+d^2)$ . Show $m*n$ can be written as sum of $2$ squares. And the same but with $4$ squares.	solution
30.6.2008	amkach	For $n \geq 2$ , 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 $(\pm1, \pm1, ..., \pm1)$ . 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
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
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
			solution

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