Áp dụng BĐT AM - GM, ta có:
a2+2b2+3
=(a2+b2)+(b2+1)+2
≥2ab+2b+2
Tương tự, ta có: b2+2c2+3≥2bc+2c+2 và c2+2a2+3≥2ac+2a+2
VT=a2+2b2+31+b2+2c2+31+c2+2a2+31
≤2ab+2b+21+2bc+2c+21+2ac+2a+21
=21(ab+b+11+bc+c+11+ac+a+11)
=21(ab+b+11+bc+c+abcabc+ac+a2bc+abcabc) (Thay abc = 1)
=21(ab+b+11+b+1+abab+1+ab+bb)
=21×ab+b+11+ab+b
=21=VP(đpcm)
Dấu "=" xảy ra khi a = b = c = 1