HMMT 十一月 2016 · GEN 赛 · 第 1 题
HMMT November 2016 — GEN Round — Problem 1
题目详情
- If a and b satisfy the equations a + = 4 and + b = , determine the product of all possible values b a 15 of ab .
解析
- If a and b satisfy the equations a + = 4 and + b = , determine the product of all possible values b a 15 of ab . Proposed by: Eshaan Nichani Answer: 1 1 1 16 1 64 2 34 We multiply a + = 4 and + b = to get 2 + ab + = ⇐⇒ ( ab ) − ( ab ) + 1 = 0. Since b a 15 ab 15 15 ( ) 2 34 − 4 · 1 · 1 > 0, there are two roots ab , so the product of both possible values of ab is 1 by Vieta’s. 14