Question

[Nguyễn Trần Đức Huy]

Cho 3 so a,b,c thoa man (a + b + c)² = 3(a² + b² + c²). Tim GTNN P = a² + (a + 2)(b + c) + 2020

Answer 1

[Nguyễn Việt Lâm]

\(\left(a+b+c\right)^2=3a^2+3b^2+3c^2\)

\(\Leftrightarrow a^2+b^2+c^2+2ab+2bc+2ca=3a^2+3b^2+3c^2\)

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}a-b=0\\b-c=0\\c-a=0\end{matrix}\right.\) \(\Leftrightarrow a=b=c\)

\(\Rightarrow P=a^2+\left(a+2\right)\left(a+a\right)+2020\)

\(\Rightarrow P=3a^2+4a+2020=3\left(a+\frac{2}{3}\right)^2+\frac{6056}{3}\ge\frac{6056}{3}\)

\(P_{min}=\frac{6056}{3}\) khi \(a=-\frac{2}{3}\)