HMMT 二月 2002 · 冲刺赛 · 第 21 题

英文解析

21. The Dyslexian alphabet consists of consonants and vowels. It so happens that afinite sequence of letters is a word in Dyslexian precisely if it alternates between consonantsand vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. Howmany letters are in the alphabet?
    Solution: 12 Suppose there are c consonants, v vowels. Then there are c · v · c · v · c +5
    2 2 2
    v · c · v · c · v = ( cv ) ( c + v ) five-letter words. Thus, c + v = 4800 / ( cv ) = 3 · (40 /cv ) , so cv isa divisor of 40. If cv ≤ 10, we have c + v ≥ 48, impossible for c, v integers; if cv = 40, thenc + 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).