MathDB
No roots but roots

Source: Tournament of Towns Spring 2016 Junior A-level

February 23, 2017
floor functionalgebraquadratics

Problem Statement

Do there exist integers aa and bb such that : (a) the equation x2+ax+b=0x^2 + ax + b = 0 has no real roots, and the equation ⌊x2⌋+ax+b=0\lfloor x^2 \rfloor + ax + b = 0 has at least one real root? (2 points) (b) the equation x2+2ax+bx^2 + 2ax + b = 0 has no real roots, and the equation ⌊x2⌋+2ax+b=0\lfloor x^2 \rfloor + 2ax + b = 0 has at least one real root? 3 points (By ⌊k⌋\lfloor k \rfloor we denote the integer part of kk, that is, the greatest integer not exceeding kk.) Alexandr Khrabrov
Back to ProblemsView on AoPS