E:15346

Soluție.

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

b) Din punctul a) putem scrie $1^{3}+1^{3}+5^{3}+6^{3}=7^{3}$ 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 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 $5^{2018}+6^{2018}<1^{2018}+1^{2018}+5^{2018}+6^{2018}<7^{2018}$, inegalitatea este demonstrată.