Japan Mathematical Olympiad

Question

For all quadruples (x, y, z, w) consisting of integers 1 x, y, z, w 1000, we consider adding up the maximum value of xy + zw, xz + yw, xw + yz and denote the sum by M. Similarly, for all quadruples (x, y, z, w) consisting of integers 1 x, y, z, w 1000, we consider adding up the minimum value of xy + zw, xz + yw, xw + yz and denote the sum by m. Determine the number of positive divisors of M - m.

Accepted Answer

20412 For 3 real numbers a, b, c, the difference of the maximum value and the minimum value of them is |a-b|+|b-c|+|c-a|2. Using |(xy + zw) - (xz + yw)| = |x - w||y - z|, we have M - m = 12 _x,y,z,w=1^1000 ( |x - w||y - z| + |x - y||z - w| + |x - z||y - w| ). Since _x,y,z,w=1^1000 |x - w||y - z| = ( _x,y=1^1000 |x - y| )^2 = ( 2 _d=1^999 d(1000 - d) )^2 = (999 1000 1001)^29, we have M - m = 32 (999 1000 1001)^29 = 2^5 3^5 5^6 7^2 11^2 13^2 37^2. Therefore, the number of positive divisors of M - m is 6 6 7 3 3 3 3 = 20412.