60th Putnam 1999 Problem A4 Let a ij = i 2 j/(3 i (j 3 i + i 3 j )). Find ∑ a ij where the sum is taken over all pairs of integers (i, j) with i, j > 0.
Let b i = i/3 i , then a ij = b i 2 b j (b i + b j ). Let the sum be k. Then k = ∑ a ij = 1/2 ∑ (a ij + a ji ) = 1/2 ∑ b i b j . We have the familiar 1 + 2x + 3x 2 + ... = 1/(1 - x) 2 , so x + 2x 2 + 3x 3 + ... = x/(1 - x) 2 , and hence ∑ b i = 1/3 (2/3) -2 = 3/4. So k = 1/2 (3/4) 2 = 9/32. 60th Putnam 1999 © John Scholes jscholes@kalva.demon.co.uk 14 Dec 1999