E:15346: Difference between revisions
Created page with "'''Soluție.''' a) <math>1^3 + 1^3 + 5^3 + 6^3 - 7^3 = 0</math>. b) Din punctul a) putem scrie <math>1^3 + 1^3 + 5^3 + 6^3 = 7^3</math> sau <math>\left(\frac{1}{7}\right)^3 + \left(\frac{1}{7}\right)^3 + \left(\frac{5}{7}\right)^3+\left(\frac{6}{7}\right)^3 = 1</math>. Acum <math>\left(\frac{1}{7}\right)^{2018}<\left(\frac{1}{7}\right)^3 , \left(\frac{5}{7}\right)^{2018}<\left(\frac{5}{7}\right)^3, \left(\frac{6}{7}\right)^{2018}<\left(\frac{6}{7}\right)^3</math>, de..." |
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'''[[E:15346]] (Mihai Vijdeluc și Vasile Ienuțaș, Baia Mare)''' | |||
''a) Calculați <math>1^3 + 1^3 + 5^3 + 6^3 - 7^3</math>. | |||
b) Arătați că <math>5^{2018}+6^{2018}<7^{2018}</math>'' | |||
'''Soluție.''' | '''Soluție.''' | ||
Latest revision as of 10:39, 13 January 2025
E:15346 (Mihai Vijdeluc și Vasile Ienuțaș, Baia Mare)
a) Calculați 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 1^3 + 1^3 + 5^3 + 6^3 - 7^3} .
b) 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 5^{2018}+6^{2018}<7^{2018}}
Soluție.
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 1^3 + 1^3 + 5^3 + 6^3 - 7^3 = 0} .
b) Din punctul a) putem scrie 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 1^3 + 1^3 + 5^3 + 6^3 = 7^3} sau 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(\frac{1}{7}\right)^3 + \left(\frac{1}{7}\right)^3 + \left(\frac{5}{7}\right)^3+\left(\frac{6}{7}\right)^3 = 1} .
Acum 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(\frac{1}{7}\right)^{2018}<\left(\frac{1}{7}\right)^3 , \left(\frac{5}{7}\right)^{2018}<\left(\frac{5}{7}\right)^3, \left(\frac{6}{7}\right)^{2018}<\left(\frac{6}{7}\right)^3} , 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 \left(\frac{1}{7}\right)^{2018}+\left(\frac{1}{7}\right)^{2018}+\left(\frac{5}{7}\right)^{2018}+\left(\frac{6}{7}\right)^{2018} < \left(\frac{1}{7}\right)^3 + \left(\frac{1}{7}\right)^3 + \left(\frac{5}{7}\right)^3 + \left(\frac{6}{7}\right)^3.}
Obținem astfel 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(\frac{1}{7}\right)^{2018}+\left(\frac{1}{7}\right)^{2018}+\left(\frac{5}{7}\right)^{2018}+\left(\frac{6}{7}\right)^{2018}<1} sau 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 1^{2018}+1^{2018}+5^{2018}+6^{2018}<7^{2018}} .
Deoarece 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^{2018}+6^{2018} < 1^{2018}+1^{2018}+5^{2018}+6^{2018} < 7^{2018} } , inegalitatea este demonstrată.