HMMT 二月 2010 · GEN2 赛 · 第 2 题

HMMT February 2010 — GEN2 Round — Problem 2

[ 3 ] The rank of a rational number q is the unique k for which q = + · · · + , where each a is the i a a 1 k 1 1 1 smallest positive integer such that q ≥ + · · · + . Let q be the largest rational number less than a 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

解析

[ 3 ] The rank of a rational number q is the unique k for which q = + · · · + , where each a is the i a a 1 k 1 1 1 smallest positive integer such that q ≥ + · · · + . Let q be the largest rational number less than a 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 1 Answer: (5 , 21 , 421) Suppose that A and B were rational numbers of rank 3 less than , and 4 1 1 1 1 1 1 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 1 of all the rationals less than with rank 3, those that have a = 5 are greater than those that have 1 4 a = 6 , 7 , 8 , . . . Therefore we can “build” q greedily, adding the largest unit fraction that keeps q less 1 1 than : 4 1 1 is the largest unit fraction less than , hence a = 5; 1 5 4 1 1 1 is the largest unit fraction less than − , hence a = 21; 2 21 4 5 1 1 1 1 is the largest unit fraction less than − − , hence a = 421. 3 421 4 5 21 1 http://en.wikipedia.org/wiki/Inscribed_angle_theorem General Test, Part 2