HMMT 二月 2018 · 冲刺赛 · 第 16 题

HMMT February 2018 — Guts Round — Problem 16

[ 9 ] Solve for x : x b x b x b x b x cccc = 122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2018, February 10, 2018 — GUTS ROUND Organization Team Team ID#

解析

[ 9 ] Solve for x : x b x b x b x b x cccc = 122 . Proposed by: John Michael Wu 122 Answer: 41 This problem can be done without needless casework. (For negative values of x , the left hand side will be negative, so we only need to consider positive values of x .) The key observation is that for x ∈ [2 , 3), 122 is an extremely large value for the expression. Indeed, we observe that: b x c = 2 b x b x cc ≤ 2(3) − 1 = 5 b x b x b x ccc ≤ 3(5) − 1 = 14 b x b x b x b x cccc ≤ 3(14) − 1 = 41 x b x b x b x b x cccc < 3(41) = 123 So the expression can only be as large as 122 if ALL of those equalities hold (the the fourth line equaling 122 40 isn’t good enough), and x = . Note that this value is extremely close to 3. We may check that 41 122 this value of x indeed works. Note that the expression is strictly increasing in x, so x = is the only 41 value that works.