HMMT 二月 2008 · 冲刺赛 · 第 20 题

HMMT February 2008 — Guts Round — Problem 20

[ 10 ] For how many ordered triples ( a, b, c ) of positive integers are the equations abc + 9 = ab + bc + ca and a + b + c = 10 satisfied?

解析

[ 10 ] For how many ordered triples ( a, b, c ) of positive integers are the equations abc + 9 = ab + bc + ca and a + b + c = 10 satisfied? Answer: 21 Subtracting the first equation from the second, we obtain 1 − a − b − c + ab + bc + ca − abc = (1 − a )(1 − b )(1 − c ) = 0. Since a , b , and c are positive integers, at least one must equal 1. Note that a = b = c = 1 is not a valid triple, so it suffices to consider the cases where exactly two or one of a, b, c are equal to 1. If a = b = 1, we obtain c = 8 and similarly for the other two cases, so this gives 3 ordered triples. If a = 1, then we need b + c = 9, which has 6 solutions for b, c 6 = 1; a similar argument for b and c gives a total of 18 such solutions. It is easy to check that all the solutions we found are actually solutions to the original equations. Adding, we find 18 + 3 = 21 total triples.