oVlad wrote: Let $a,b,c,d$ be positive real numbers satisfying $a^2+b^2+c^2+d^2=1$. Prove that \[a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.\] After homogenisation and full expanding we need to prove that:$$\sum_{sym}(a-b)^2(3a^2b^2+3c^4+3d^4+5c^2d^2-2abcd)\geq0,$$which is obvious