HMMT 二月 2002 · 冲刺赛 · 第 45 题
HMMT February 2002 — Guts Round — Problem 45
题目详情
- [9] Find the number of sequences a , a , . . . , a of positive integers with the property 1 2 10 that a = a + a for n = 1 , 2 , . . . , 8, and a = 2002. n +2 n +1 n 10
解析
- Find the number of sequences a , a , . . . , a of positive integers with the property 1 2 10 that a = a + a for n = 1 , 2 , . . . , 8, and a = 2002. n +2 n +1 n 10 Solution: 3 Let a = a, a = b ; we successively compute a = a + b ; a = a + 1 2 3 4 2 b ; . . . ; a = 21 a + 34 b . The equation 2002 = 21 a + 34 b has three positive integer 10 solutions, namely (84 , 7) , (50 , 28) , (16 , 49), and each of these gives a unique sequence.