Gazeta matematică 2015: Difference between revisions
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== Gazeta Matematică 9/2015 == | == Gazeta Matematică 9/2015 == | ||
'''[[E:14892]] (Radu Pop & Ienuțaș Vasile)''' | |||
''Fie triunghiul'' <math>ABC</math> ''cu'' <math>m\left(\sphericalangle C\right) > 30^\circ</math> ''și punctele'' <math>M</math>, <math>P</math>, <math>R</math>, <math>T</math>. ''Punctul'' <math>M</math> ''este situat în interiorul triunghiului'' <math>ABC</math> ''astfel încât'' <math>m\left(\sphericalangle BMA\right) = 120^\circ</math> ''și <math>m\left(\sphericalangle BCM\right) = 30^\circ</math>, punctul <math>P\in \left(MD\right.</math> astfel încât <math>\left[MP\right] \equiv \left[MB\right]</math> cu <math>AM \cap BC = \left\{D\right\}</math>, iar <math>R\in \left(AB\right)</math> și <math>T \in \left(AC\right)</math> astfel încât <math>m\left(\sphericalangle RBM\right) = \frac{1}{2} \cdot m\left(\sphericalangle RPM\right)</math> și <math>m\left(\sphericalangle TPM\right) = 2 \cdot m\left(\sphericalangle TCM\right)</math>.'' | |||
# ''Arătați că'' <math>\frac{1}{2} \cdot m\left(\sphericalangle RPT\right) = m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MTR\right)</math> | |||
# ''Determinați măsura unghiului'' <math>\sphericalangle ARM</math> | |||
# ''Arătați că'' <math> m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MAT\right) = m\left(\sphericalangle DMC\right)</math> | |||
Revision as of 08:26, 1 December 2024
Gazeta Matematică 1/2015
27020 (Gheorghe Szöllösy)
Să se calculeze suma 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 \sum_{k=0}^{\left[\frac{n}{2}\right]} \frac{1}{4^k \cdot (k!)^2 (n-2k)!}, \quad n \geq 1. }
27022 (Guntter Gotha)
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 f:\left[a,b\right] \to \mathbb{R}} o funcție cu proprietatea lui Darboux și 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 f\left(a\right) \cdot f\left( b \right) >0} . Mulțimea 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 M = \left\{ x \in \left[ a, b \right] \, | \, f\left(x\right) =0 \right\}} este finită și are un număr impar de elemente. Demonstrați 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 f} are un punct de extrem local ce aparține mulțimii 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 M} .
27024 (Gheorghe Szöllösy)
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 I_n = \int_{0}^{\pi} \frac{\cos nx}{13-12\cos x}\,dx, n\ge0.} 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}(I_0+I_1+I_2+\ldots+I_n).}
Gazeta Matematică 2/2015
27036 (Radu Pop)
Să se determine funcțiile derivabile 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 f : \mathbb{R} \to \mathbb{R} } cu proprietățile:
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 f' } este funcție strict crescătoare;
b) 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 f'(0) = 0; }
c) 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 f(yf'(x)) + f(x)f(y) = xy f'(x)f'(y) } , oricare ar fi 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,y \in \mathbb{R} } .
Gazeta Matematică 3/2015
Gazeta Matematică 9/2015
E:14892 (Radu Pop & Ienuțaș Vasile)
Fie triunghiul 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 ABC} 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 m\left(\sphericalangle C\right) > 30^\circ} și punctele 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 M} , 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 P} , 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 R} , 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 T} . Punctul 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 M} este situat în interiorul triunghiului 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 ABC} astfel încât 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 m\left(\sphericalangle BMA\right) = 120^\circ} 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 m\left(\sphericalangle BCM\right) = 30^\circ} , punctul 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 P\in \left(MD\right.} astfel încât 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[MP\right] \equiv \left[MB\right]} 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 AM \cap BC = \left\{D\right\}} , iar 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 R\in \left(AB\right)} 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 T \in \left(AC\right)} astfel încât 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 m\left(\sphericalangle RBM\right) = \frac{1}{2} \cdot m\left(\sphericalangle RPM\right)} 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 m\left(\sphericalangle TPM\right) = 2 \cdot m\left(\sphericalangle TCM\right)} .
- 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 \frac{1}{2} \cdot m\left(\sphericalangle RPT\right) = m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MTR\right)}
- Determinați măsura unghiului 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 \sphericalangle ARM}
- 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 m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MAT\right) = m\left(\sphericalangle DMC\right)}