S:E15.325: Difference between revisions

Latest revision as of 18:46, 3 December 2024

E:15325 (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:

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

Soluție:

Prin ridicare la pătrat 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 \frac{d + 2 \cdot \overline{abc}}{\overline{d0}^2} = \frac{3}{\overline{abc}}}

Notăm 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} = x} și avem:

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 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 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 = 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} .

@@ Line 1: / Line 1: @@
-'''S:E15.325 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)'''
+'''E:15325 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)'''
-''Determinațti numerele naturale   \( \overline{abcd} \) pentru care ..''
+''Determinați numerele naturale <math>\overline{abcd}</math> pentru care:''
+<math display="block">\frac{\sqrt{d+2 \cdot \overline{abc}}}{\overline{d0}} = \frac{\sqrt{3 \cdot \overline{abc}}}{\overline{abc}}</math>
 '''Soluție:'''
-''''
+''Prin ridicare la pătrat obținem:''
+<math display="block">\frac{d + 2 \cdot \overline{abc}}{\overline{d0}^2} = \frac{3}{\overline{abc}}</math>
+''Notăm <math>\overline{abc} = x</math> și avem:''
+<math display="block">x \cdot (d + 2x) = 300d^2</math>
+''sau''
+<math display="block">2x^2 + dx - 300d^2 = 0</math>
+''cu soluțiile:''
+<math display="block">x = 12d \quad \text{și} \quad x = -\frac{25d}{2}</math>
+''Prin urmare <math>\overline{abc} = 12d</math>. Pentru <math>d = 9</math> obținem <math>\overline{abc} = 108</math> și numărul căutat este <math>1089</math>, iar pentru <math>d \leq 8</math> nu avem soluții deoarece <math>12d \leq 96</math>.''