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. 
. Still, if you don't, somebody else will; all additions are appreciated.
| Added | Author | Problem | Solutions
|
|---|---|---|---|
| 2012-10-? | parigi | Can you arrange ten points in the plane in such a way that they cannot be covered by a set of non-overlapping unit disks? | solution by zwim |
| 2012-10-08 | Niall | Four point frogs initially form a square in the plane. Any frog can jump over any other frog in such a way that the other frog forms the midpoint of its jump. Can the frogs ever form a larger square? Can any three of them ever be colinear? | solution by gu1729 |
| 2012-10-08 | 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. | solution by YogSothoth |
| 2012-08-02 | Warfreak2 | Without reference to any external material, prove that the side:diagonal ratio of a regular pentagon is 2 : 1+√5. | solution by flyingdario |
| 2012-08-02 | maproom | Find a set of sets of natural numbers such that the sum of all the numbers is 20 and their product is as great as possible.
(This is a trick question. This trick is in understanding the properties of sets.) |
solution by weiqidevil |
| 2012-06 | maproom | A, B and C have a cycle race from E to F. All three set out at 10am at different speeds from E. A is 5km/h faster than B and 10km/h faster than C. They all maintain a constant speed over the course except as follows. Also travelling the same route at a constant speed is a burger van. Any cyclist reaching the van immediately loses a constant 20km/h in speed. The van leaves E at 9am and arrives in F at 3pm; the race itself is a three-way tie. When does it finish? | solution by DanielTom |
| 2012-03-05 | maproom | 31 points are equally spaced in a circle. In how many distinct ways can you pick 6 of them such that every pair of picked points is at a different separation? | solution by iceweasel |
| 2012-01-20 | twillo | You have 64 coins heads up and 36 coins tails up - you have to split them into 2 piles with equal numbers of heads in each. You are permitted to turn coins over but can never tell what state they are in. | |
| 2012-01-25 | maproom | Find a permutation group G acting transitively on N letters, and a permutation group H also acting transitively on N letters, such that G and H are isomorphic, but no isomorphism between them corresponds to any mapping of the two sets of letters. | solution by D239500800 |
| 2012-01-20 | twillo | An odd number of points are arranged in the plane with no three colinear. Prove that for each of these points, the number of triangles (whose vertices are others of the points) within which it lies is even. | solution by Warfreak2 |
| 2011-11-21 | maproom | You deal a standard bridge pack to four players in the usual way. Which is more likely, and by roughly how much:
(a) You have cards in only two suits, or (b) There is some suit in which both you and your partner have no cards? |
solution by Zahlman |
| 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. |
solution |
| 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. | Unsolved |
| 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 |
