HMMT 十一月 2014 · 冲刺赛 · 第 9 题

HMMT November 2014 — Guts Round — Problem 9

[ 7 ] Let f be a function from the nonnegative integers to the positive reals such that f ( x + y ) = f ( x ) · f ( y ) holds for all nonnegative integers x and y . If f (19) = 524288 k , find f (4) in terms of k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND Organization Team Team ID#

解析

[ 7 ] Let f be a function from the nonnegative integers to the positive reals such that f ( x + y ) = f ( x ) · f ( y ) holds for all nonnegative integers x and y . If f (19) = 524288 k , find f (4) in terms of k . 4 / 19 n Answer: 16 k The given condition implies f ( mn ) = f ( m ) , so 19 4 f (4) = f (4 · 19) = f (19 · 4) = f (19) 4 / 19 and it follows that f (4) = 16 k .