Gazeta matematică 2014: Difference between revisions

Latest revision as of 08:18, 19 January 2025

Gazeta Matematică 1/2014

E:14682 (Cristina Vijdeluc și Mihai Vijdeluc)

Se consideră 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} în care $m(\angle A)=2\cdot m(\angle B)+30^{\circ }$ . 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 M} este situat pe segmentul 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 (BC)} 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 AM = AC} . 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 m(\angle MAC) = 2 \cdot m(\angle MAB)} , 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 BM = MC} .

Gazeta Matematică 5/2014

26927 (Radu Pop și Vasile Ienuțaș)

Polinomul 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 = ax^3 + bx^2 + cx + d\in \mathbb{R}\left[X\right]} are toate rădăcinile reale și verifică inegalitatea 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 - 4ac < ad - 4a^2} . Să se arate că rădăcinile nu pot fi toate strict pozitive.

Gazeta Matematică 11/2014

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} , 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 c} avemFailed 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 + b| + |a + c| \ge |b - c|.} b) Demonstrați că pentru orice număr real 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} avemFailed 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| + |x + 2| + |x + 3| + ... + |x + 2014| \ge 1007^2.}

Gazeta Matematică 12/2014

E:14763 (Cristina Vijdeluc și Mihai Vijdeluc)

Determinați numerele prime 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} 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 q} , 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 p > q} , știind 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 p\left( 1+3pq\right) + q\left(1-3pq\right) = p^3 - q^3} .

@@ Line 4: / Line 4: @@
 ''Se consideră triunghiul <math>ABC</math> în care <math>m(\angle A) = 2 \cdot m(\angle B) + 30^\circ</math>. Punctul <math>M</math> este situat pe segmentul <math>(BC)</math> astfel încât <math>AM = AC</math>. Dacă <math>m(\angle MAC) = 2 \cdot m(\angle MAB)</math>, arătați că <math>BM = MC</math>.''
+== Gazeta Matematică 5/2014 ==
+'''[[26927]] (Radu Pop și Vasile Ienuțaș)'''
+''Polinomul <math>f = ax^3 + bx^2 + cx + d\in \mathbb{R}\left[X\right]</math> are toate rădăcinile reale și verifică inegalitatea <math>b^2 - 4ac < ad - 4a^2</math>. Să se arate că rădăcinile nu pot fi toate strict pozitive.''
 == Gazeta Matematică 11/2014 ==
@@ Line 10: / Line 16: @@
 ''a) Arătați că oricare ar fi numerele reale <math>a</math>, <math>b</math>, <math>c</math> avem''<math display="block">|a + b| + |a + c| \ge |b - c|.</math>''b) Demonstrați că pentru orice număr real <math>x</math> avem''<math display="block">|x + 1| + |x + 2| + |x + 3| + ... + |x + 2014| \ge 1007^2.</math>
+== Gazeta Matematică 12/2014 ==
+'''[[E:14763]] (Cristina Vijdeluc și Mihai Vijdeluc)'''
+''Determinați numerele prime <math>p</math> și <math>q</math>, cu <math>p > q</math>, știind că <math>p\left( 1+3pq\right) + q\left(1-3pq\right) = p^3 - q^3</math>.