MathDB

1

Part of 2016 Vietnam National Olympiad

Problems(2)

A system of equations

Source: Vietnam Mathematical Olympiad 2016, Day 1, Problem 1

1/6/2016
Solve the system of equations {6x−y+z2=3x2−y2−2z=−1emsp;emsp;(x,y,z∈R.)6x2−3y2−y−2z2=0\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1   (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}.
algebra proposedalgebrasystem of equations
VMO 2016 Problem 5

Source: VMO 2016

5/6/2016
Find all a∈Ra\in\mathbb{R} such that there is function f:R→Rf:\mathbb{R}\to\mathbb{R} i) f(1)=2016f(1)=2016 ii) f(x+y+f(y))=f(x)+ay \forall x,y\in\mathbb{R}
algebrafunctional equationfunction