Vì a,b,c \in [0,1]
\Rightarrow b^2\leq b, c^3\leq c
\Rightarrow a+b^2+c^3-ab-bc-ca\leq a+b+c-ab-bc-ca
Mặt khác, ta có: abc-ab-bc-ca+a+b+c-1=(a-1)(b-1)(c-1)
\Rightarrow a+b+c-ab-bc-ca=(a-1)(b-1)(c-1)-abc+1\leq 1
Suy ra \Rightarrow a+b^2+c^3-ab-bc-ca\leq 1 (đpcm)
Dấu "=" xảy ra \Leftrightarrow...