363
2197. Proposed by Joaqu n Gomez Rey, IES Luis Bu~nuel, Alcorc on, Madrid, Spain. Let n be a positive integer. Evaluate the sum: 1 X
,2k k
: 2k+1 k=n (k + 1)2
2198. Proposed by Vedula N. Murty, Andhra University, Visakhapatnam, India. Prove that, if a, b, c are the lengths of the sides of a triangle, 2 1 2 1 2 1 2 2 (b , c) bc , a2 + (c , a) ca , b2 + (a , b) ab , c2 0; with equality if and only if a = b = c. 2
2199. Proposed by David Doster, Choate Rosemary Hall, Wallingford, Connecticut, USA. Find the maximum value of c for which (x + y + z )2 > cxz for all 0 x < y < z. 2200. Proposed by Jeremy T. Bradley, Bristol, UK and Christopher J. Bradley, Clifton College, Bristol, UK. Find distinct positive integers a, b, c, d, w, x, y , z , such that z2 , y2 = x2 , c2 = w2 , b2 = d2 , a2
and
c2 , a2 = y2 , w2:
Bonus Problem for 1996 220A?. Proposed by Ji Chen, Ningbo University, China. Let P be a point in the interior of the triangle ABC , and let 1 = \PAB, 1 = \PBC , 1 = \PCA. p Prove or disprove that 3 =6. 1 1 1