E:16407

16407 (Cristina Vijdeluc, Salonic și Mihai Vijdeluc)

Aflați cifrele nenule 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 } 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 b} pentru care 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 + 10 \cdot (a + b)^{3} = \overline{baba}.}

Soluție:

Egalitatea din enunț se scrie $a+10\cdot (a+b)^{3}={\overline {bab0}}+a,$ de unde $10\cdot (a+b)^{3}={\overline {bab0}}$ și apoi $(a+b)^{3}={\overline {bab}}.$ Căutând prin încercări cuburi perfecte de 3 cifre, găsim $7^{3}=343$. 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 a = 4} 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 b = 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 \overline{ab} = 43.}