Vol. 10 No 3 - Pi Mu Epsilon

PROBLEMS AND SOLUTIONS 24 1then xy' - alx = axy + bgY, ory' = c + ay + bxy,subject to the conditions y(0) = 0 and y' (0) = c. To solve this system wemake the substitution y = u exp(ax + b212) and we getu = c exp(-ax - b&) with u(0) = 0 and u'(0) = c.xFrom this equation we obtain the solution u = c f exp(-at - bt 2 /2) dt and0r 1 r 1842. [Fall 19941 Proposed by Russell Euler, Northwest Missouri StateUniversity, Mary ville, Missouri.Let xi be a positive real number for i = 1, 2, . . . , n. Prove that- -with equality if and only if xl = x2 = ... = x,,.Solution by Joe Howard, New Mexico Highlands University, Las Vegas,New Mexico.By the Cauchy-Schwartz inequality we have thatWe observe that the integral exists for all x and hence that the given seriesconverges. Since S = y(l), we obtains = c exp [(a 1: [- $1 exp (a dtor equivalently,andCombining (1) and (2) we get thatIf b > 0, then by making the substitution a + bt = uJb, we canobtainwhere $(-r) is the cumulative function of the normal probability distribution,defined bya .Also solved by Murray S. Klamkin, and the Proposer.and the theorem follows. It is easy to see that we have equality if and onlyifxl = x2 = ... = xn.Also solved by Miguel Amengual Covas, Seung-Jin Bang, Scott H.Brown, Paul S. Bruckman, Philip A. D. Castoro, William Chau, RichardI. Hess, Murray S. Klamkin, Henry S. Lieberman, David E. Manes, Can.A. Minh, Yoshinobu Murayoshi, Bob Prielipp, St. Olaf Problem SolvingGroup, Selvaratnam Sridharma, Sammy Yu and Jimmy Yu, and theProposer.Klarnkin showed more generally that