Vol. 8 No 7 - Pi Mu Epsilon

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So&t-tton by Hmy Sedingm and ChffAZe~ R. VXmiiLe, St. Bonauen-tuAe UyU.uu~.-ity, St. BonauentuAe, New Yoik. Choose point P on BC such that 4BDP = 3BDA. Then, by ASA, triangles ABD and PBD are congruent. Therefore DF = AD = DE. We see that 4ADB = ^CDE = 60'. Then ^CDF = 60Â° too. Thus, by SAS, triangles CDP and CDE are congruent and we get that 4BEC = WEC = 120' - WE = 120' - W F = 120' - 4ACB= 80'. SotiLtLon by Jack Gcui.6unkel, Ftiiihing, New Yo&. Since AE and AC are tangent to the circle, then OA bisects ^CAE. Similarly OC bisects 3ACF. Hence 1 +OAC = -^CAE 1 = -(180Â 2 - ~BAC) = 90' - ~ 2 B A C Ah0 tiotved by RUSSELL EULER, NoAthtuut HtcAhouA^ State UiM.ue~&Xy, MmyvJUULe, JACK GARFUNKEL, F&tAh&g, NY, ROBERT C. GEBHARDT, Hopatcong, NJ, RICHARD A. GIBBS, Po& Lcitf-ci Co-fcZege, VuAango, CO, PETER GEISER, St. Ctoud State Uni.uu~.-c-ty, MN, RICHARD I. HESS, Rancho Paloh Vvidu, CA, RALPH KING, St. BonaventuAe Un-CumJULy, NY, JOHN D. MOORES, Cambtu.dge, MA, NORTHWEST MISSOURI STATE UNIVERSITY MATHEMATICS CLUB, MmyvWLe, JOHN H. SCOTT, MueoJLutm CoU.ege, S h t Pad, MN, WADE H. SHERARD, Fman UH-tvmJut.y, GieenuJUULe, SC, ARTHUR H. SIMONSON, E u t Texm State. UVIA.uvu.-ity a t Texo~kana, KENNETH M. WILKE, Topeka, KS, and the PROPOSER. Some of the tiotuu~. u~ed the taw of, hhu. GLEN E. MILLS, Uiange County Public Sehoot&, OnJLando, FL, iiihg decimal udue~, found the. angLe to withiin 0.05'. One 0 t h ~ ~ bubm-cibion uhumed that D -LA the. midpoint of AC and a 6Lnd papa wad in appiying the. ims of thu, each obtaining a Wiong anhum. 638. [Fall 19861 Piopotied by R. S. LuthaM., UnLuWs^ty of Whcoiui.n Cintm, JanuuWLe, Whco&n. In the figure on the next page, the circle with center 0 is an excircle of triangle ABC. OA is produced to meet BK in D. quadrilateral. Then BK is drawn so that ^KBA = bloc, and Prove that OCBD is a cyclic and similarly Hence ~AOC = 1800 - (WAC + ~ACO) Now we have that W C t W C = W A + W C 1 wo = go0 - $BCA. 1 1 = ~ B A C + ~ B C A = 9 0 - +AN. + 3AOC = 4ABC + 23AOC = 3ABC + 180' - 4ABC = 180Â° so quadrilateral DBCO is cyclic, since a pair of opposite angles are supplementary. SAWJULLVL bo.euAc.ow4 w ~ubm-c.fcted by RICHARD A. GIBBS, FoA-t Leuici CoUege, Vmgo, CO, RALPH KING, St. BonuventuA.e Ufu.vmJULy, NY, HENRY S. LIEBERMAN, Waban, MA, JOHN D. MOORES, Camblu.dge, MA, NORTHWEST MISSOURI STATE UNIVERSITY MATHEMATICS CLUB, MoA~WLC, JOHN H. SCOTT, Macuhtm CoUege, Saint Pad, MN, MADE H. SHERARD, Fman Ufu.uuu,m, GieenvWLe, SC, ARTHUR H. SIMONSON, E u t Texa S-tote UyU.vi~^ltq at TexaAkana, and the PROPOSER. L& In Memoriam ~auve taught a backbreaking load of trigonometry and algebra at Algonquin College in Ottawa, Ontario, Canada. Nevertheless, to keep from becoming mathematically stale, in 1975 he founded a small problem journal called Eureka, later renamed Crux Mathematicom, "a
puzzle or problem for mathematicians." Singlehandedly he was soon editing 10 issues a year, each 30 pages in length, a monumental task for any person. Rarely could you find an error on its pages, so careful was its editor. He personally verified every mathematical statement that appeared on its 300 pages each year. His wit and fluency in both French and English made the journal lively and exciting, as well as informative. Early in 1986 L ~O retired because of ill health, after having built Crux into a journal with subscribers and contributors throughout the world. He died in June, 1987. Therefore it is our privilege to dedicate this issue of the Problem Department to the memory of ~ 6o ~auv6, a true scholar and friend, and to express our hope that we will always be guided by his spirit in our editing of these pages. THE RICHARD V. ANDREE AWARDS Richard V. Andree, Professor Emeritus of the University of Oklahoma, died on May 8, 1987, at the age of 67. Professor Andree was a Past-President of Pi Mu Epsilon. He also served the fraternity as Secretary-Treasurer General and as Editor of the Pi Mu Epsilon Journal. At the summer meetings in Salt Lake City the fraternity Council voted to designate the prizes in the national student paper competition as Richard V. Andree Awards. First prize winner for 1986-1987 is Wah Keung Chan, McGi11 University, for his paper "On the Largest RAT-free Subset of a Finite Set of Points." Wah's paper appeared in the Spring 1987 issue. Wah will receive $200. Second prize winner is Jennifer Zobitz, College of St. Benedict, for her paper "Fractals: Mathematical Monsters.'' Jennifer's paper is the lead article in this issue of the Journal. Jennifer will receive $ 100. Third prize winner is Kelly Ann Chambers, University of Dayton, for her paper "The Isomorphism of the Lattice of Congruence Relations on a Group and the Lattice of Normal Subgroups of a Group." Kelly Ann's paper also appears in this issue of the Journal. Kelly Ann will receive $50. Congratulations Wah, Jennifer and Kelly Ann. 1987 NATIONAL PI MU EPSILON MEETING The National Meeting of the Pi Mu Epsilon Fraternity was held at the University of Utah in Salt Lake City on August 5 through August 8. Highlights included a reception for students, faculty advisors, and alum- ni, a Council Luncheon and business meeting, the Annual Banquet, and in- formal student parties. The J. Sutherland Frame Lecturer was Professor Clayton W. Dodge, editor of the Journal's Problem Department. He enter- tained his listeners with "Reflections of a Problem Editor." The program of student papers included: Deterministic and Probabili~tic Fire Models The Epidemiology of the AIDS Virus The Strangely Attracted Bouncing Ball W e n E. &totAde^Â Md~b~cbe-fctAf-pha Kto~ce^-te/i. Po$gtech& Irea-fct^tute Amon Kiebano66 CaJLHofww. Lambda. Ufitumity 06 Co^i.60Ai'Lta, %viA Stephafite Ruth Land Texu Lambda Un-LumUy 06 Texu Applications of Signal ProcessingVeb~ Shale Mai&achtJb&itti VeJLta ufitvmity 06 Low& Self-calibration of CmpZex Visi- AÂ£ Sa6a.d-vuJLi. bility Data from a Very Large Array Iowa AQfea of Antennas Iowa State. Ivu.umity Representations and Characters of Groups An Algebraic Construction of a Projective Geometry Ken Chick Ohm Vet.& M-tflnn ufitvmity 9kphaM-t.e. Uumo~lu. Cati{lo~wUn Thita Ocu.dwtai CoUege The RSA Public Key Cpyptosystem: Stephen Fie-te An Application for Modem Algebra W U V-tAfliVu.0. Af-pk W u t V-tAfliWASH Ufitvmitg 16/64 = 1/4 and Other ii-digit David A. Me^bheo Canoe flations C0nnec-tLCu-t B&fa uvu.vm.ity 06 HaAt^oid
Page 2 and 3: JOURNAL - VOLUME 8 NUMBER 7 Fractal
Page 4 and 5: measure yield varying results. Henc
Page 6 and 7: Figure 2 From THE FRACTAL GEOMETRY
Page 8 and 9: where N represented the number of p
Page 10 and 11: computer. Diversity of graphic imag
Page 12 and 13: medians concur and that the altitud
Page 14 and 15: So by Ceva's Theorem for Chords, AA
Page 16 and 17: THE ISOMORPHISM OF THE LATTICE OF C
Page 18 and 19: - (a,b) E f(Nl V N2) (a,b) E f(S1N2
Page 20 and 21: If f(x) = A(x) is a polynomial of d
Page 22 and 23: PUZZLE SECTION Edited by Jobeph V .
Page 24 and 25: Whacnohtic No. 25 The 302 letters t
Page 26 and 27: 657. Pmpo~ed by R. S. LtLtha~~, Uni
Page 28 and 29: BowLing Gheen, Kentucky. t by 8m.q
Page 30 and 31: for n = 0, 1, 2, ... and k a given
Page 32 and 33: SIMONSON, Ean,! Tern State UnivmJUL
Page 36 and 37: An Algebraist's View of Competitive

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