37th Putnam 1976

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Problem A2

Let a(x, y) be the polynomial x2y + xy2, and b(x, y) the polynomial x2 + xy + y2. Prove that we can find a polynomial pn(a, b) which is identically equal to (x + y)n + (-1)n (xn + yn). For example, p4(a, b) = 2b2.

 

Solution

Let us write E(n) = (x + y)n + (-1)n (xn + yn). We use induction on n. We have E(1) = 0, E(2) = 2b, E(3) = 3a.

We find that (x + y)n+3 = b (x + y)n+1 + a (x + y)n and xn+3 + yn+3 = b (xn+1 + yn+1) - a (xn + yn). So it follows that E(n+3) = b E(n+1) + a E(n). That completes the induction.

 


 

37th Putnam 1976

© John Scholes
jscholes@kalva.demon.co.uk
23 Jan 2001