Gazeta matematică 2013: Difference between revisions
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''Se consideră șirul de numere reale <math>(x_n)_{n \geq 0}</math> și <math>(y_n)_{n \geq 0}</math> cu <math>x_n \geq 1</math>, <math>y_n \geq 1</math>, pentru orice <math>n \in \mathbb{N}</math>, și <math>\lim_{{n \to \infty}} (x_n^2 + y_n^2) = 2</math>. Să se calculeze <math>\lim_{{n \to \infty}} x_n</math> și <math>\lim_{{n \to \infty}} y_n</math>.'' | ''Se consideră șirul de numere reale <math>(x_n)_{n \geq 0}</math> și <math>(y_n)_{n \geq 0}</math> cu <math>x_n \geq 1</math>, <math>y_n \geq 1</math>, pentru orice <math>n \in \mathbb{N}</math>, și <math>\lim_{{n \to \infty}} (x_n^2 + y_n^2) = 2</math>. Să se calculeze <math>\lim_{{n \to \infty}} x_n</math> și <math>\lim_{{n \to \infty}} y_n</math>.'' | ||
== Gazeta Matematică 6-7-8/2013 == | |||
'''[[E:14527|14527]] (Cristina Vijdeluc şi Mihai Vijdeluc)''' | |||
''Pentru orice număr natural nenul <math>n</math> , notăm <math>n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n</math> și <math>0! = 1</math>.'' | |||
''a) Arătați că <math>\left( n+1\right) \cdot \bigl(n+1\bigr) ! - n \cdot n ! = \left( n^2 + n + 1 \right) \cdot n ! </math>'' | |||
''b) Dacă <math>A = \left( 60^2 + 60 + 1\right) \cdot 60 ! + \left( 59^2 + 59 + 1\right) \cdot 59! + \ldots + \left( 1^2 + 1 + 1\right) \cdot 1! + (0^2 + 0 + 1) \cdot 0!</math>, atunci <math>A</math> se divide cu <math>2013^2</math>.'' | |||
Revision as of 12:47, 2 January 2025
Gazeta Matematică 1/2013
E:14440 (Vasile Ienuțaș și Radu Pop)
Se consideră numărul natural 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 a_1,a_2,a_3,.....,a_{2012}} sunt numere prime, mai mari sau egale cu 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 5} . Arătați 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 B=2 \cdot A + 2013 } nu este pătrat perfect.
26713 (Radu Pop și Vasile Ienuțaș)
Se consideră șirul de numere 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 (x_n)_{n \geq 0}} și 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_n)_{n \geq 0}} cu 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_n \geq 1} , 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_n \geq 1} , pentru orice 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}} , și 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 \lim_{{n \to \infty}} (x_n^2 + y_n^2) = 2} . Să se calculeze 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 \lim_{{n \to \infty}} x_n} și 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 \lim_{{n \to \infty}} y_n} .
Gazeta Matematică 6-7-8/2013
14527 (Cristina Vijdeluc şi Mihai Vijdeluc)
Pentru orice număr natural nenul 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} , notăm 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 \cdot 2 \cdot 3 \cdot \ldots \cdot n} și 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 0! = 1} .
a) Arătați 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 \left( n+1\right) \cdot \bigl(n+1\bigr) ! - n \cdot n ! = \left( n^2 + n + 1 \right) \cdot n ! }
b) Dacă 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 = \left( 60^2 + 60 + 1\right) \cdot 60 ! + \left( 59^2 + 59 + 1\right) \cdot 59! + \ldots + \left( 1^2 + 1 + 1\right) \cdot 1! + (0^2 + 0 + 1) \cdot 0!} , atunci 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} se divide cu 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 2013^2} .