Vol. 5 No 10 - Pi Mu Epsilon

"Whconhin.324. Ptopohed by R. S. Lut\m, U~uwmLtq 06 [ U ~ C O I I . ~ ~ IJanc~wLUe,I ,Evaluate325. Ptopohed by Cl~utLu (0. Th,igg, Salt Viego, Cdi6ot1ua.Show that there is only one third-order magic square xith positiveprime elements and a magic constant of 267.Solutions292. [Spring 19731 Ptopobed by Jack Gut6u1tk& Fohuf H ~ Y A HighSchool, Fhlung, Nu Yotk.If perpendiculars are ccnstruc-ed at the points of tanFen(:y cf theincircle gf a triangle and extended outward to equal length:,then theioins of the% endpoints form a triangle perspective with the given tri-angle.SoLCLon by Chyfon W. Dodge, U n i w ~ d 06 y !.faine at Oho110.FIGURE 3the incircle touch sides BC, CA, AB of triangle ABC at pointsX, Y, Z, (see Fig. 3) and let the other ends of the constructed equalperpendiculars be P, Q, R, respectively.It is well known that pointsA, B, C and X, Y, Z are in perspective at the Gergonne point for trian-gle ABC. But also triangles XYZ and PQR are in perspective at point I,the incenter for triangle-ABC.It is also known that perspective istransitive; that since triangles ABC and XYZ are in perspective and tri-angles XYZ and PQR are in perspective, then triangles ABC and PQR arein perspective .also.A.00 holwed by ZAZOU KATZ, Bewdy H i L h , Cdi6ohnia, and fhePtopoh WL.Ptoblm EcLLtotlh NoZeTWO recent texts describing the Gergonne point and other notablepoints of the triangle are:Rinehart and Ninston, Inc., 1969; (2) Clayton W.(1) David C. Kay, Cozzege Geometry, Holt,Dodge, EucZidean Geome-try and Transformations, Addison-Wesley Publishing Company, Inc., 1972.293. [Spring 19731 Phopohed by Lau Kot~mLi, !.10hgafl s m e College,W n o t e , hfmy&znd.Prove that N = 531° + is divisible by 78.1. SoLutio~t by R. Robimon Roiue, Saucamento, Cdi6otlua.Since 53 * -l(mod 6) and 103 : l(mod 6), we have 53lU3 E (-1)lo3(mod 6): -l(mo6 6) and 1 l(mod 6).Alsc, since 53 : l(mod 13) and 103 : -l(mod 131, we have 531° !l(mod 13) and 1 0 3 5 ~ -l(mod ~ 13).Hence N is divisible by both 6 and 13, which are relatively prime.Therefore fi is divisible by 6 . 13 = 7B.S.i.m&RAYMOND E.h o W o ~ u wme odimed by MERYL J. ALTABET, &onx, N. Y.;ANDERSON? !.lo~tta~ta State UniwmiAy, Bozana~~, !.lo&na; DONALDCASCI, Rltodt Zh.f!.und College, Phowidence, Rl~ode Zhhnd; RICHARD A. GIBBS,Foa Laud College, Quha~tgo, Colomdo; DONNELLY JOHNSON, hlajoh, USAF,A h Fohce ZnhLLtuLe 06 Tech~~oLogy, Fahbot11, Oluo; BRUCE .LOVETT yRuLg~College N u Bhu~t~wick, N. J.; T. E. MOORE, LVLL~~~LU&VL SXuAe College,Wdg watm, !.lab.; THERESA PRATT, N. Eafon, blab. ; PAOLO RANALDI ,Aho~, Oluo; MICHAEL SCHWARZSCHILD , PoLyfechnic 71%+fi't&te o 6 BhookLyn.11. SoLCLon by C l M u (0. Th,igg, Sun Viego, Cdi6otnia.Examining the congruences of 531° and 1 0 3 for ~ ~ the moduli 3, I+,