HMMT 二月 1998 · ADV 赛 · 第 8 题

HMMT February 1998 — ADV Round — Problem 8

Given any two positive real numbers x and y , then x y ◊ is a positive real number defined in terms of x and y by some fixed rule. Suppose the operation x y ◊ satisfies the equations ( x ⋅ y ) ◊ y = x y y ( ◊ ) and ( x ◊ 1 ) ◊ x = x ◊ 1 for all x y , > 0 . Given that 1 ◊ 1 = 1 , find 19 ◊ 98 .

解析

Given any two positive real numbers x and y , then x y is a positive real number defined in terms of x and y by some fixed rule. Suppose the operation x y satisfies the equations ( x · y ) y = x ( y y ) and ( x 1) x = x 1 for all x , y > 0. Given that 1 1 = 1, find 19 98. Answer: 19 . Note first that x 1 = ( x · 1) 1 = x · (1 1) = x · 1 = x. Also, x x = ( x 1) x = x 1 = x. 19 19 19 Now, we have ( x · y ) y = x · ( y y ) = x · y. So 19 98 = ( · 98) 98 = · (98 98) = · 98 = 19 . 98 98 98