MathDB

Problem 3 of Second round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

August 31, 2019

algebraFunctional Equationsfunctions

Problem Statement

Let

f: \mathbb{R}^2\rightarrow \mathbb{R}

be a function for which for arbitrary

x,y,z\in \mathbb{R}

we have that

f(x,y)+f(y,z)+f(z,x)=0

. Prove that there exist function

g:\mathbb{R}\rightarrow \mathbb{R}

for which:

f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}