Japan Mathematical Olympiad

Question

For positive real numbers x, y, the positive real number x y is defined as x y = xxy+1. Calculate the following expression: ((( (((100 99) 98) 97) ) 3) 2) 1.

Accepted Answer

100495001 For any positive real numbers x, y, and z, the following equation holds: (x y) z = x y(x y)z + 1 = xxy+1xxy+1z + 1 = xxy + xz + 1 = xx(y+z) + 1 = x (y+z). Using this repeatedly, we have: aligned & ((((((100 99) 98) 97) 96) 3) 2) 1 &= ((((((100 (99 + 98)) 97) 96) 3) 2) 1 &= ((((100 (99 + 98 + 97)) 96) 3) 2) 1 &= &= 100 (99 + 98 + 97 + + 3 + 2 + 1) &= 100 99 1002. aligned Therefore, the answer is 100 4950 = 100495001.