HMMT 二月 2010 · 代数 · 第 2 题

英文解析

2. [ 3 ] The rank of a rational number q is the unique k for which q = + · · · + , where each a is theia a
   1 k
   1 1 1
   smallest positive integer such that q ≥ + · · · + . Let q be the largest rational number less thana a 4
   1 i
   1 1 1
   with rank 3, and suppose the expression for q is + + . Find the ordered triple ( a , a , a ).
   1 2 3
   a a a
   1 2 3
   Answer: (5 , 21 , 421) Suppose that A and B were rational numbers of rank 3 less than , and 1
   1 1 1 1 1 14
   let a , a , a , b , b , b be positive integers so that A = + + and B = + + are the
   1 2 3 1 2 3
   a a a b b b
   1 2 3 1 2 3
   1 1
   expressions for A and B as stated in the problem. If b < a then A < ≤ < B . In other words,
   1 1
   a − 1 b
   1 1
   of all the rationals less than with rank 3, those that have a = 5 are greater than those that have 1
   a = 6 , 7 , 8 , . . . Therefore we can “build” q greedily, adding the largest unit fraction that keeps q less 41
   than :11
   1 14
   is the largest unit fraction less than , hence a = 5;
   5 41
   1 1 1
   is the largest unit fraction less than − , hence a = 21;
   21 4 52
   1 1 1 1
   is the largest unit fraction less than − − , hence a = 421.
   421 4 5 213