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HMMT 十一月 2016 · GEN 赛 · 第 1 题

HMMT November 2016 — GEN Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. If a and b satisfy the equations a + = 4 and + b = , determine the product of all possible valuesb a 15
    of ab .
解析

英文解析

  1. If a and b satisfy the equations a + = 4 and + b = , determine the product of all possible valuesb 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. Sinceb a 15 ab 15 15
    ( ) − 4 · 1 · 1 > 0, there are two roots ab , so the product of both possible values of ab is 1 by Vieta’s.342
    14