Vol. 10 No 5 - Pi Mu Epsilon

PROBLEM DEPARTMENTEdited by Clayton W. DodgeUniversity of MaineThis department welcomes problems believed to be new and at a levelappropriate for the readers of this journal. Old problems displaying noveland elegant methods of solution are also invited. Proposals should beaccompanied by solutions if available and by any information that will assistthe editor. An asterisk (*) preceding a problem number indicates that theproposer did not submit a solution.All communications should be addressed to C. TV. Dodge, 5752Neville/Math, University of Maine, Orono, ME 04469-5752. E-mail:dodge@ganss.nmemat.maine.edu. Please submit each proposal and solutionpreferably typed or clearly written on a separate sheet (one side only)properly identified with name and address. Solutions to problems in thisissue should be mailed to arrive by July 1, 1997.Problems for Solution888. Proposed by the Editor.In 1953 Howard Eves' book An Introduction to the History ofMathematics was first published. It quickly became the definitiveundergraduate text in mathematics history. It still is today. To honor thisoutstanding text and its equally outstanding author, solve this base ninealphameric, finding the unique value of H EWMATH + HIST = HEVES.889. Proposed by M. S. Klamkin, University of Alberta, Edmonton,Alberta, Canada.Prove thatwherea> 1 andx> y> 0.x Y 2890. Proposed by Peter A. Lindstrom, Irving, Texas.Express the following sum in closed form, where real number a # 1:n891. Proposed by John Wahl, Aft. Pocono, Pennsylvania, and AndrewCusumano, Great Neck, New York.Solve for d the equationbcd + cda + dab + abc = E~,a+b+c+d892. Proposed by Bill Correll, Jr., student, Denison University,Granville, Ohio.Prove that the average of the eigenvalues of a real, symmetric,idempotent matrix is at most one.893. Proposed by Peter A. Lindstrom, Irving, Texas.Show that the sequence {xn} converges and find its limit, where x, = 2and, for n >. 1,894. Proposed by Andrew Cusumano, Great Neck, New York.Let us lake P, = 4/2fi, P, = 8 { T f i , p4 =16/m/2, and so forth. Find the value of lim n+ n(Pn - P,,).895. Proposed by Andrew Cusumano, Great Neck, New York.Let ABC be an isosceles right triangle with right angle at C. Erectsquares ACEH and ABDI outwardly on side AC and hypotenuse AB. Let CImeet BH at 0 and AB at K, and let A0 meet BC at J. Let DE cut AB at Fand AC at G. It is known (Problem 817, Fall 1994, page 72) that DE passesthrough 0. Let JF meet AH at S and let JG meet BH at T. Finally, let BHand AC meet at M and let JM and CI meet at L. See the figure.a) Prove thati) ST is parallel to DOE,