E:15346

E:15346 (Mihai Vijdeluc și Vasile Ienuțaș, Baia Mare)

a) Calculați $1^{3}+1^{3}+5^{3}+6^{3}-7^{3}$.

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) $1^{3}+1^{3}+5^{3}+6^{3}-7^{3}=0$.

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 $\left({\frac {1}{7}}\right)^{3}+\left({\frac {1}{7}}\right)^{3}+\left({\frac {5}{7}}\right)^{3}+\left({\frac {6}{7}}\right)^{3}=1$.

Acum $\left({\frac {1}{7}}\right)^{2018}<\left({\frac {1}{7}}\right)^{3},\left({\frac {5}{7}}\right)^{2018}<\left({\frac {5}{7}}\right)^{3},\left({\frac {6}{7}}\right)^{2018}<\left({\frac {6}{7}}\right)^{3}$, 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 $\left({\frac {1}{7}}\right)^{2018}+\left({\frac {1}{7}}\right)^{2018}+\left({\frac {5}{7}}\right)^{2018}+\left({\frac {6}{7}}\right)^{2018}<1$ 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 $5^{2018}+6^{2018}<1^{2018}+1^{2018}+5^{2018}+6^{2018}<7^{2018}$, inegalitatea este demonstrată.