Gazeta matematică 2014

Gazeta Matematică 11/2013

E:14742 (Liliana Puț)

a) Arătați că oricare ar fi numerele reale 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 a} , 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 b} , $c$ avem

$$|a+b|+|a+c|\geq |b-c|.$$

b) Demonstrați că pentru orice număr real $x$ avem

$$|x+1|+|x+2|+|x+3|+...+|x+2014|\geq 1007^{2}.$$