S:E15.325

S:E15.325 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)

Determinați numerele naturale 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{abcd}} pentru care:

$${\frac {\sqrt {d+2\cdot {\overline {abc}}}}{\overline {d0}}}={\frac {\sqrt {3\cdot {\overline {abc}}}}{\overline {abc}}}$$

Soluție:

Prin ridicare la pătrat obținem:

$${\frac {d+2\cdot {\overline {abc}}}{{\overline {d0}}^{2}}}={\frac {3}{\overline {abc}}}$$

Notăm ${\overline {abc}}=x$ și avem:

$$x\cdot (d+2x)=300d^{2}$$

sau

$$2x^{2}+dx-300d^{2}=0$$

cu soluțiile:

$$x=12d\quad {\text{și}}\quad x=-{\frac {25d}{2}}$$

Prin urmare ${\overline {abc}}=12d$. Pentru $d=9$ obținem 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{abc} = 108} și numărul căutat este 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 1089} , iar pentru 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 d \leq 8} nu avem soluții 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 12d \leq 96} .