E:16407

16407 (Cristina Vijdeluc, Salonic și Mihai Vijdeluc)

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

Soluție:

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