mmmwin.jpg

The results are in, and the winners for MMM #6 are . . . Henno Brandsma and Steven King! Hope you guys enjoy your gift certificate to The Art of Problem Solving! The turnout for this contest wasn’t nearly as good as the previous one (48 submissions compared to 103), so that tells us that we need to do a bette job advertising and spreading the word. It also tells us that we need a better/more interesting math problem.

For the next contest, we will probably revert back to a single problem. Originally, I thought the idea of having an easier problem and a harder problem would be more appealing to people. Perhaps we’ll try to do a double problem again next week! Any feedback would be appreciated!

Once again, we would like to thank our sponsors from The Art of Problem Solving! Stay tuned for the next Monday Math Madness!

Part I Explained

As explained by Efrit Freeq

Each step dilutes the fraction of mayonnaise to 500/510 of what it was at the
previous step. It starts at 100% mayonnaise and we need to dilute it to 50%.

So we need n such that (500/510)^n ~= 1/2

Hence n ~= ln(1/2)/ln(500/510) so n = 35 is as close as we can get.

Part 2 Explained

Explanation by Lieven Marchand

Let us denote by p(n) the number of ways Nortrom can eat n pounds of
beef. Since he will always start by eating either a 1 pound burger or
a two pound burger, p(n) obeys the following recursion:

p(n)=p(n-1)+p(n-2) with initial conditions p(1)=1 and p(2)=2.

These are the well known Fibonacci numbers and so the solution is

p(n)=1/sqrt(5) [((1+sqrt(5))/2)^(n+1) - ((1-sqrt(5))/2)^(n+1)]

This gives us p(17)=2584 and p(25)=121393.