Vol. 5 No 2 - Pi Mu Epsilon

Solution by Murray S. Klamkin, Ford Scientific Laboratory.It follows by symmetry thatTotal for all 36 values of M 801For each M, N the corresponding MI, N' necessarily also appearsin the tabulation.B) If (abc)(def) = uvwxyz and (cba)(fed) = zyxwvu, then (c) (f) = pn,where n = p t 1. The only possible terminal duos are 2(6) = 12,3(4) = 12, 5(9) = 45, and 7(8) = 56.Now in PI, 399(499) = 199101, so 3. 4 may not be a terminal duo.Also, 299(699) = 209001, but (b2)(e6) = lOObe + 20(e + 3b) + 12,so in P the penultimate digit is not zero, which rules out 2,6 as a terminal duo.If (5b5)(9e9) or (7b7)(8e8) provide a solution, the P = PI, sothe product must be palindromic and therefore divisible by 11Hence, any solutions must come from (5b5)(979), (7b7)(858),or (737)(8e8). There are only four such solutions:where c = fi.We first transform the rectangular coordinates (x,y) into ellipticcoordinates (E,n) (see Stratton, Electromagnetic Theory, McGraw-Hill, N.Y., 1941, pp. 52-54.)Herer tr1 2£EÑ 2c. n=- 1-22cand the region of integration is c > 1, -1 < n < 1.AlsoThus,2 EZ-,,~ 2 z 112dxdy = c { - a -1 dcdn.€2 1-nzNo other solutions appear when the products (5b9)(9e5), (7b8)(8e7),(5b7)(9e8), and (5b8)(9e7) are exhausted.11. Solution by Carl A. Argila, TRW Systems, Houston, Texas.Given any three digit integer, I, we define the function B asfollows :Integrating with respect to n;Now let 6 = cosh0 giving-where [A] is the greatest integer in A. Note that B(1) is justI written backwards. We wish to find all pairs of three digitintegers, M and N, for with B(M) and B(N) are also three digitintegers and for whichB(HxN) = B(tt)xB(N).By means of a simple computer program we determine that thereare 805 distinct pairs of three digit numbers which satisfy thiscondition.83. (Spring 1956) Proposed by G.K. Horton, University of Alberta.EvaluateDifferentiating the known integralmKo(a) = f e-a cOshOdO (KO - modified Bessel function)0twice with respect to a, we obtain1K = ~ ( a )0a1 0Whence,m- -K (a) = f cosh20 e* c0sh8de.