Vol. 5 No 10 - Pi Mu Epsilon

which is greater:- 12 tan-'(fi - 1) or 3 tan-'(l/~) + tan (5'99) ?31 7. Phopobed by tke E&oh 06 tke PhobLem Depa,ttinent.A rectangle ADEB is constructed externally on the hypctenuse AB ofa right triangle ABC (Fig. 1). The lines CD and CE inter~ect the lineAB in the points F and G respectively. a) If DE = AD^, show thatA G ~ + F B ~ = AB~. b) If AD = DE, show that F G ~ = AF GB.320. Paopv~cd by H. S. !.I. Cvxcteh, Tctoi~tv, Ca~lada.Prove that the projectivity ABCX BCD (for 4 collinear points) isof the period 4 if and only if H(AC, BD).321 . Pavpobcd by Nob~nv King, RdcLgh, Nv~th Cc~~fi~ta. (Dedicatedto .t/~e tnctnvag 06 Lco Lfvbm)According to bierten's Theorem.where y denotes Euler's constant, (9.57721 ... ) an^ where the product onthe left is taken over all primes not exceeding n, (See Hardy and Wight,The Theory of Numbers, p, 351 01- Trygve Nagellis Introduction to hnberTheory, p. 298).Can you estimateYoak,322. Paopobed bg Jack Gm~unke.L!, Fokut Hi.L& High School, NCWIt is known that the ratio of the perimeter of a triangle to thesum of its altitudes is gpeater than or equal to 2/&.,Vathematical i.IonthZy, $roblem E lk27, 1961, pp. 296-297).(See AmericanProve thestronger inequality for the internal angle bisectors t t and t :a' bFIGURE 131 8. Paopob ed by R. Ro biiaon Rome, Sacmne~do, Cafi~vaiua.Two equal cylindrical tanks, Tank A abcve Tank B, have equal ori-fices in their fl:>ors, capable of discharging water at the rate of 13figallons per minute, where h is the depth of water in feet.A t 10:2,, a.m.Tank B is empty and water 1s 19 feet deep in Tank A, as discharge begins.At noon Tank A is Just emptied, What was the maximum depth in Tank B,and when ?How deep is the water in Tank B at noon, and when will itbe srqty?-319. Paopobed by Pao6e~boa 1.1. S. LongueA-Uiggi~a, Catnb~dge,England.Let A', B', C ibe the images of an arbitrary pcint in the ?idesBC, CA, AB of a triangle ABC. Prove that the 4 circles AB':', BC1t!',CA'B' , ABC are all concurrent.equality holding if and only if the triangle is equilateral.323. Paopvbed by David L. Sdhmnan, Lob Angeh , CaLL6omua.Call plane curves such as the circle of radius 2, the square ofside 4? or the 6 x 3 rectangle in Fig, 2 bornetrfc if their perimeterts numerically equal to the area they enclose. \!hat is the maximum areathat can be enclosed by an isometric curve?FIGURE 2