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We claim that setting u = and u = yields, respectively, f (N 1) 0. For the first claim, note that the first two terms inside sum to less than -(n2 + n)-1, and the rest is less than u4 C 0 u2, which produces a negative total when n > 5. For the second claim, when n > 8 the first two terms suffice (that is, u3/3 > 1/n), while for 5 5 the local minima occur at [(n + N/3n + 1 1)/21 and at [(n /3n I + 1)/2].

10750 [1999, 686]. Proposed by Leonard Smiley, University of Alaska, Anchorage, AK. For a positive integer m, express En??= (n/ gcd(m, n))x" as a rational function of x. Solution I by David Callan, University of Wisconsin, Madison WI. For each positive integer d, let V(d) = Hpd (1 p), where the product is over all primes p dividing d. Let A(x) be the desired generating function, and let B(x) = Ld1n, V(d)xd/(1 _ Xd)2. We prove that A(x) = B(x). Since xd/(i xd)2 = Z1:0 kxkd, the coefficient of xn in the power series expansion of B(x) is