27401

\usepackage{MnSymbol}

27401 (Radu Pop, Baia Mare)

Fie $n\in \mathbb {N}$. Să se arate că

$$(n+1)ab(a+b)-(4n^{3}+13n+10)ab+(n+2)^{3}(a+b)\geq (n+2)^{3},$$

oricare ar fi $a,b\in [1,\infty )$

Soluție:

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

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 \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) \ge}

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 \ge (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 =}

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

Rezultă 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)(x+1)(y+1)(x+y+n^2+n) \ge (n+2)^3xy} . 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 x=a-1\ge 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} .