27401

27401 (Radu Pop)

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} . Să se arate că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+1)ab(a+b)-(4n^3+13n+10)ab+(n+2)^3(a+b) \geq (n+2)^3, } oricare ar fi $a,b\in [1,\infty )$

Soluție:

Fie $x,y\in [0,\infty )$. Avem

$$(x+1)(y+1)(x+y+n^{2}+n)=(n+1){\biggl (}\underbrace {{\frac {x}{n+1}}+{\frac {x}{n+1}}+\ldots +{\frac {x}{n+1}}} _{(n+1){\text{ ori}}}+1{\biggr )}\cdot$$

$$\cdot {\biggl (}\underbrace {{\frac {y}{n+1}}+{\frac {y}{n+1}}+\cdots +{\frac {y}{n+1}}} _{(n+1){\text{ ori}}}+1{\biggr )}\cdot {\biggl (}{\frac {x}{n+1}}+{\frac {y}{n+1}}+\underbrace {1+1+\cdots +1} _{n{\text{ ori}}}{\biggr )}\geq$$

$$\geq (n+1){\sqrt[{n+2}]{\frac {x^{n+1}}{(n+1)^{n+1}}}}\cdot {\sqrt[{n+2}]{\frac {y^{n+1}}{(n+1)^{n+1}}}}\cdot {\sqrt[{n+2}]{\frac {xy}{(n+1)^{2}}}}\cdot (n+2)^{3}=$$

$$={\frac {xy}{n+1}}(n+2)^{3}.$$

Rezultă că

$$(n+1)(x+1)(y+1)(x+y+n^{2}+n)\geq (n+2)^{3}xy.$$

Fie $x=a-1\geq 0$ şi Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=b-1\ge0} .

Obţinem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+1)ab(a+b+n^2+n-2)\ge(n+2)^3(a-1)(b-1),} de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+1)ab(a+b)+(n^3+2n^2-n-2)ab \ge (n+2)^3ab-(n+2)^3(a+b)+(n+2)^3} deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n+1)ab(a+b)-(4n^2+13n+10)ab+(n+2)^3(a+b) \ge (n+2)^3.}