Bạn chỉ cần CM 1 số bất đẳng thức phụ là :[tex]a^{4}+b^{4}+c^{4}\geq a^{2}bc+b^{2}ac+c^{2}ab[/tex]
=>[tex]\frac{1}{a^{4}+b^{4}+c^{4}+abcd}+\frac{1}{b^{4}+c^{4}+d^{4}+abcd}+\frac{1}{c^{4}+d^{4}+a^{4}+abcd}+\frac{1}{a^{4}+b^{4}+d^{4}+abcd}\leq \frac{1}{a^{2}bc+b^{2}ac+c^{2}ab+abcd}+\frac{1}{b^{2}cd+c^{2}bd+d^{2}bc+abcd}+\frac{1}{a^{2}cd+c^{2}ad+d^{2}ac+abcd}+\frac{1}{a^{2}bd+b^{2}ad+d^{2}ab+abcd}=\frac{1}{abc(a+b+c+d)}+\frac{1}{bcd(a+b+c+d)}=\frac{1}{acd(a+b+c+d)}+\frac{1}{abd(a+b+c+d)}=\frac{(a+b+c+d)}{abcd(a+b+c+d)}=\frac{1}{abcd}[/tex] (đpcm)