2010 IMO Shortlist A2

Let the real numbers $a, b, c, d$ satisfy the relations $a+b+c+d=6$ and $a^{2}+b^{2}+c^{2}+d^{2}=12$. Prove that $$36 \leq 4\left(a^{3}+b^{3}+c^{3}+d^{3}\right)-\left(a^{4}+b^{4}+c^{4}+d^{4}\right) \leq 48$$ (Ukraine)

令实数$a、b、c、d$满足关系$a+b+c+d=6$和$a^{2}+b^{2}+c^{2}+d^{2}=12$。证明 $$36 \leq 4\left(a^{3}+b^{3}+c^{3}+d^{3}\right)-\left(a^{4}+b^{4}+c^{4}+d^{4}\right) \leq 48$$ (乌克兰)