S:E15.325: Difference between revisions
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'''Soluție:''' | '''Soluție:''' | ||
''Prin ridicare la pătrat obținem | ''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>.'' | |||
Revision as of 07:29, 1 December 2024
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:
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:
Notăm și avem:
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 2x^2 + dx - 300d^2 = 0}
cu soluțiile:
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 = 12d \quad \text{și} \quad x = -\frac{25d}{2}} Prin urmare 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} = 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} .