sandu2508 wrote: Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Choose for example any $x\in\mathbb R$ and : $a=\left(\frac{50x^4}{x^4+1+(x+1)^4}\right)^{\frac 14}$ $b=\left(\frac{50}{x^4+1+(x+1)^4}\right)^{\frac 14}$ $c=-a-b$