This section is not just for problems and solution for Putnam 2008, but all stuff related to our second week.
By the way, now you should feel free to see solutions of the Putnam 2009 problems, especially those which you
tried and were not successful.
A1 is neat; Please don't read this unless you have tried A1.
ReplyDeleteIt is given that f(x, y) + f(y, z) + f(z, x) = 0
for all x, y, and z. What happens if you set x=y=z? You will get f(x, x) = 0 for all x.
Now what happens if you set z=x. Then you can conclude that f(x, y) = -f(y, x) why? because in the resulting equation f(x, x) = 0. Now we can simply define g(x) = -f(0, x) and set z=0 in the given equation. That does it. Please check.
Did anyone get a chance to look at A2? It is a nice 2-player game with determinants. Consider two players, say A and B, who take turns in filling the 2008 x 2008 matrix with arbitrary numbers. Suppose A goes first. A wins if the determinant of the resulting matrix (after it is completely filled) is non-zero, otherwise B wins. The question is to find if there is a winning strategy for either player. I claim that B has a winning strategy. Here is my solution. Think of an imaginary vertical line that splits the matrix into equal parts. Now since A goes first, B will simply mimick the entries of A by placing them in the corresponding cell on the opposite side of the vertical line! At the end, the entries on both sides of vertical line are identical, a fortiori, the determinant is zero. This is one winning strategy for B. After solving this problem I looked into the solutions and I was terribly impressed by another solution by Manjul Bhargava's. He came up with a strategy for B in which the sum of entries in any row will be zero and consequently the determinant will be zero. How is that possible? Very easy (only after knowing the solution!) if A puts an x, then B would put a -x in the same row Clearly all this would be possible only if the square matrix has even order. What would happen if it has an odd order. Something to think about it..
ReplyDeleteTalking about games and winning strategies here is another interesting game. Consider an empty circle in which two players A and B take turns in placing pennies. A penny that is being placed cannot overlap an existing penny. A player losses if he cannot place a penny that does not overlap an existing penny. As before, assume that A goes first. Who has a winning strategy?
ReplyDeleteThese are some good games to entertain your friends and family at a party.
I have my blog on the roll. Check it out:
ReplyDeletehttp://sunilchebolu.wordpress.com/
You may find the ISBN game interesting.
How is Putnam problem solving going? There has been no activity on the blog this week. I guess we are all very busy.