HMMT 二月 2002 · 冲刺赛 · 第 21 题

HMMT February 2002 — Guts Round — Problem 21

[7] The Dyslexian alphabet consists of consonants and vowels. It so happens that a finite sequence of letters is a word in Dyslexian precisely if it alternates between consonants and vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. How many letters are in the alphabet?

解析

The Dyslexian alphabet consists of consonants and vowels. It so happens that a finite sequence of letters is a word in Dyslexian precisely if it alternates between consonants and vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. How many letters are in the alphabet? 5 Solution: 12 Suppose there are c consonants, v vowels. Then there are c · v · c · v · c + 2 2 2 v · c · v · c · v = ( cv ) ( c + v ) five-letter words. Thus, c + v = 4800 / ( cv ) = 3 · (40 /cv ) , so cv is a divisor of 40. If cv ≤ 10, we have c + v ≥ 48, impossible for c, v integers; if cv = 40, then c + v = 3 which is again impossible. So cv = 20, giving c + v = 12, the answer. As a check, this does have integer solutions: ( c, v ) = (2 , 10) or (10 , 2).