Question

Xét các số thực dương x,y thoả mãn điều kiện x³ + y³ +(x + y)³ + 30xy = 2000. Chứng minh rằng x +y =10

Answer 1

\(\Leftrightarrow\left(x+y\right)^3-3xy\left(x+y\right)+\left(x+y\right)^3+30xy=2000\)

\(\Leftrightarrow2\left[\left(x+y\right)^3-1000\right]-3xy\left(x+y-10\right)=0\)

\(\Leftrightarrow2\left(x+y-10\right)\left[\left(x+y\right)^2-10\left(x+y\right)+100\right]-3xy\left(x+y-10\right)=0\)

\(\Leftrightarrow\left(x+y-10\right)\left[2\left(x+y\right)^2-20\left(x+y\right)+200-3xy\right]=0\)

\(\Leftrightarrow x+y=10\)

Do:

\(2\left(x+y\right)^2-20\left(x+y\right)+200-3xy\)

\(=\left(x+y-10\right)^2+\left(x+y\right)^2-3xy+100\)

\(=\left(x+y-10\right)^2+\left(x+\dfrac{y}{2}\right)^2+\dfrac{3y^2}{4}+100>0\)