MathDB

1

Part of 2009 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Real roots of quadratics imply real roots of other quadratic

Source: XVIII Olimpíada Matemática Rioplatense (2009)

7/23/2011
Find all pairs (a,b)(a, b) of real numbers with the following property: Given any real numbers cc and dd, if both of the equations x2+ax+1=cx^2+ax+1=c and x2+bx+1=dx^2+bx+1=d have real roots, then the equation x2+(a+b)x+1=cdx^2+(a+b)x+1=cd has real roots.
quadraticsalgebra unsolvedalgebra
Functional equation: f(xy)=max{f(x+y),f(x)f(y)}

Source: XVIII Olimpíada Matemática Rioplatense (2009)

7/23/2011
Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that f(xy)=max⁡{f(x+y),f(x)f(y)}f(xy)=\max\{f(x+y),f(x) f(y)\} for all real numbers xx and yy.
functionalgebradomainalgebra unsolved