Question

[Angela jolie]

Cho ba số a,b,c thỏa mãn a+b+c=0 và |a| ≤ 1, |b| ≤ 1, |c| ≤ 1. Chứng minh rằng a⁴+b⁶+c⁸ ≤ 2

Answer 1

[Nguyễn Việt Lâm]

Do \(-1\le a;b;c\le1\)

\(\Rightarrow\left(1-a\right)\left(1-b\right)\left(1-c\right)+\left(1+a\right)\left(1+b\right)\left(1+c\right)\ge0\)

\(\Leftrightarrow1-abc-a-b-c+ab+bc+ca+1+abc+b+c+c+ab+bc+ca\ge0\)

\(\Leftrightarrow2\left(ab+bc+ca\right)+2\ge0\)

\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)+2\ge a^2+b^2+c^2\)

\(\Leftrightarrow\left(a+b+c\right)^2+2\ge a^2+b^2+c^2\)

\(\Leftrightarrow a^2+b^2+c^2\le2\)

Mà \(\left|a\right|;\left|b\right|;\left|c\right|\le1\Rightarrow\left\{{}\begin{matrix}a^4\le a^2\\b^6\le b^2\\c^8\le c^2\end{matrix}\right.\)

\(\Rightarrow a^4+b^6+c^8\le a^2+b^2+c^2\le2\)

Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(-1;0;1\right)\) và các hoán vị