Question

1) tìm tất cả bộ 3 số dương x,y ,z thỏa mãn ?

2x²⁰¹⁶=y¹⁵+z¹⁵; 2y²⁰¹⁶=z¹⁵+x¹⁵;  2z²⁰¹⁶=x¹⁵+y¹⁵

2)  cho x+y+xy=1. tìm giá trị nhỏ nhất của P= \(\frac{1}{x+y}+\frac{1}{x}+\frac{1}{y}\)

Answer 1

1/

Đề \(\Rightarrow z^{15}+x^{15}-\left(y^{15}+z^{15}\right)=2\left(y^{2016}-x^{2016}\right)\)

\(\Rightarrow x^{15}-y^{15}=2\left(y^{2016}-x^{2016}\right)\)

+Nếu \(x=y\text{ thì }VT=0=VP\)

+Nếu \(x>y\text{ thì }VT>0>VP\)

+Nếu \(x

Answer 2

\(1=x+y+xy\le x+y+\frac{\left(x+y\right)^2}{4}=\left(\frac{x+y}{2}+1\right)^2-1\)

\(\Rightarrow\left(\frac{x+y}{2}+1\right)^2\ge2\Rightarrow\frac{x+y}{2}+1\ge\sqrt{2}\Rightarrow x+y\ge2\sqrt{2}-2\)

\(1=x+y+xy\ge2\sqrt{xy}+xy=\left(\sqrt{xy}+1\right)^2-1\)

\(\Rightarrow\left(\sqrt{xy}+1\right)^2\le2\Rightarrow\sqrt{xy}+1\le\sqrt{2}\Rightarrow\sqrt{xy}\le\sqrt{2}-1\)

\(\Rightarrow xy\le3-2\sqrt{2}\)

\(P=\frac{1}{x+y}+\frac{1}{x}+\frac{1}{y}=\frac{x+y+xy}{x+y}+\frac{x+y}{xy}\)

\(=1+\left(\frac{xy}{x+y}+\frac{\left(\sqrt{2}-1\right)^2}{4}.\frac{x+y}{xy}\right)+\frac{1+2\sqrt{2}}{4}.\frac{x+y}{xy}\)

\(\ge1+2\sqrt{\frac{xy}{x+y}.\frac{\left(\sqrt{2}-1\right)^2}{4}\frac{x+y}{xy}}+\frac{1+2\sqrt{2}}{4}.\frac{2\sqrt{2}-2}{3-2\sqrt{2}}=\frac{5+5\sqrt{2}}{2}\)

Dấu bằng xảy ra khi và chỉ khi \(x=y=\sqrt{2}-1\)