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        riddles >> putnam exam (pure math) >> Couple of 2003 Questions
        
(Message started by: THUDandBLUNDER on Dec 7^th, 2003, 4:56am)

Title: Couple of 2003 Questions
Post by THUDandBLUNDER on Dec 7^th, 2003, 4:56am

1) Do there exist polynomials a(x), b(x), c(y), and d(y) such that
1 + xy + x²y² [smiley=equiv.gif] a(x)c(y) + b(x)d(y)?

2) Let A, B, C be equidistant points on the circumference of a circle of unit radius centred at O, and let P be any point in the circle's interior. Let a, b, c be the distances from P to A, B, C, respectively. Show that there is a triangle with side lengths a, b, c and that the area of this triangle depends only on the distance from P to O.

Title: Re: Couple of 2003 Questions
Post by Eigenray on Dec 7^th, 2003, 10:58pm

This years problems have already been posted to sci.math (http://groups.google.com/groups?selm=bquk0q%2418m%241%40news.math.niu.edu). Extremely simple solutions to all the problems, which make you feel like a moron for not getting them, will also appear shortly. I always try to avoid reading them until I can solve them myself.

Damn it, I thought B1 was supposed to be easy, to make you feel better after doing so horribly on part A. I still haven't gotten this one.

For B5, I [hide]let the points on the circle be 1, w, w²=w', where w is a primitive cube root of 1 (z' = z-bar), then plugged a=|p-1|, b=|p-w|, c=|p-w'| into Hero's formula, K=sqrt[s(s-a)(s-b)(s-c)], which expands to
16K² = 2(a²b²+a²c²+b²c²)-(a⁴+b⁴+c⁴),
and then did about half a page of algebra to get K = (1-|p|²)sqrt(3)/4. It's not too bad if you make some clever substitutions and keep using that w+w'=-1, ww' = 1[/hide].

A1 was easy, A2 was neat. A3 is on this site somewhere, and now I see an infinitely simpler proof than mine already on sci.math. B2 and B3 aren't very hard, and I haven't really thought much about the others yet.