Summation

USA AIME 1991 1 Find x2 + y2 if x and y are positive integers such that xy + x+ y = 71x2y + xy2= 880. (0) Rectangle ABCD has sides AB of length 4 and CB of length 3. Divide AB into 168 congruent segments with points A = P0, P1, . . . , P168 = B, and divide CB into 168 congruent segments with points C = Q0, Q1, . . . , Q168 = B. For 1 ? k ? 167, draw the segments PkQk. Repeat this construction on the sides AD and CD, and then draw the diagonal AC. Find the sum of the lengths of the 335 parallel segments drawn. Expanding (1 + 0.2)1000 by the binomial theorem and doing no further manipulation gives ( 1000 0 ) (0.2)0 + ( 1000 1 ) (0.2)1 + ( 1000 2 ) (0.2)2 + ? ? ?+ ( 1000 1000 ) (0.2)1000 = A0 +A1 +A2 + ? ? ?+A1000, (0) where Ak = (1000 k )