# Mathematical Sciences Research Institute

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# Serious Math

Over the years, I feel like I’ve come to know you — your political leanings and life experiences, your writing style, sense of humor and average snark level. But what about your math skills?

For example: Can you (or any high school student you know) do this?

Show that there are only finitely many triples (x, y, z) of positive integers satisfying the equation abc = 2009(a + b + c).

Or this?

Let n be an integer greater than 3. Points V1, V2, …, Vn, with no three collinear, lie on a plane. Some of the segments ViVj , with 1 *< i < j < n, are constructed. Points Vi and Vj are neighbors if ViVj is constructed. Initially, chess pieces C1,C2, …,Cn are placed at points V1, V2, …, Vn (not necessarily in that order) with exactly one piece at each point. In a move, one can choose some of the n chess pieces, and simultaneously relocate each of the chosen piece from its current position to one of its neighboring positions such that after the move, exactly one chess piece is at each point and no two chess pieces have exchanged their positions. A set of constructed segments is called harmonic if for any initial positions of the chess pieces, each chess piece Ci(1< i < n) is at the point Vi after a finite number of moves. Determine the minimum number of segments in a harmonic set.

(*Note: This sign (Cynthia Day, a Lynbrook High junior from San Jose, Lynnelle Ye, a Palo Alto High School grad bound for Stanford University, and Jing Jing Li, a recent graduate of Cupertino High from Sunnyvale, who’s going to UC Berkeley.

The team, sponsored by the Mathematical Sciences Research Institute in Berkeley, will compete next week against other brilliant mathematical minds in Shijiazhuang, China. They’re already overseas.

I think they’ll be blogging, too. Here’s a link:http://www.msri.org/specials/gmo/2010