## A challenge / plea to math folks

Two interesting math contest related articles this week. First in the Atlantic an article about different participation rates in math contests for girls and for boys:

Math for Girls, Math for Boys

By the way, the 2009 study by Ellison and Swanson mentioned in the article has one of the most interesting (and frankly baffling) statements about math education that I’ve ever seen. From the top of page 3:

“Whereas the boys come from a variety of backgrounds, the top-scoring girls
are almost exclusively drawn from a remarkably small set of super-elite schools: as many girls come from the top 20 AMC schools as from all other high schools in the U.S. combined.”

As far as I know, no one has done a follow up study to see what’s going on at those 20 schools. Seems like a very interesting topic to study.

The second article was about the US team finishing 2nd at the European Girls’ Math Olympiad:

US Team Takes Second at European Girls’ Mathematical Olympiad

After reading this article I clicked through to see the problems, and problem #1 caught my attention:

I found the problem to be fun to solve and also pretty interesting mathematically. It struck me as a great problem to use to show how mathematicians think.

Tim Gowers did an fantastic “live blog” showing his own thought process in solving a problem from the International Mathematics Olympiad a few years ago:

Tim Gowers walks through an IMO problem

I love this self-depricating line: “You idiot Gowers, read the question: the a_n have to be positive integers.”

Richard Rusczyk also has a fantastic example showing mathematical thinking involved in solving a contest problem – in this case problem #24 from the 2013 AMC 12.

So, my challenge to math folks is this – live blog your solution to problem #1 from the 2016 European Girls’ Mathematical Olympiad. Share your thinking, share your false starts, share the “aha” moments. I think this problem provides a great opportunity for people to see how people in math think, and, importantly, that the path to the solution of a problem isn’t always a straight line.

1. either I misread the text of the problem (which usually is the case) or it is a joke…

2. suevanhattum,

What can x sub n+1 (not part of the set used) = x sub 1 mean?

• Sorry if this doesn’t get typeset quite right . . . .

What the problem wants you to show is that the minimum of the sum of the squares of two consecutive terms is less than the max of twice the product of two consecutive terms.

The condition that x_{n+1} = x_{1} is there for convenience, I guess, and is just saying that for purposes of the min / max condition the term after x_n is x_1.

I thought about the numbers being arranged in a circle when I was thinking about the problem.

3. suevanhattum,

So the n numbers involved can be anything >= 0, and that condition is basically meaningless, except as a hint about how you might want to proceed?

• I think the sequence 10, 9, 10 shows that the condition isn’t meaningless. If you don’t go around in a circle there are only two pairs to consider in the condition of the problem, and the numbers in those pairs are the same. the sum of squares is 181 and twice the product is 180. So, without the “looping around in a circle” condition (since I don’t know what else to call it!) what the problem is asking you to prove wouldn’t be true.