E:16379: Difference between revisions

Revision as of 18:44, 27 December 2023

E:16379 (Cristina Vijdeluc, Salonic şi Mihai Vijdeluc, Baia Mare)

Aflaţi numărul natural ${\overline {ab}}$ , cu cifre distincte, pentru care $({\overline {ab}}-{\overline {ba}}):(a-b)={\overline {bb}}\cdot {\overline {ba}}-2015.$

Soluție:

Egalitatea din enunţ se mai scrie ${\overline {ab}}-{\overline {ba}}=(a-b)\cdot ({\overline {bb}}\cdot {\overline {ba}}-2015).$ Descompunând 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}} şi 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{ba}} în sistemul zecimal şi efectuând câteva calcule, obţinem: 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 9(a - b) = (a - b) \cdot (\overline{bb} \cdot \overline{ba} - 2015).}

Cum 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 \ne 0,} egalitatea devine 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{bb} \cdot \overline{ba} - 2015 = 9,} adică 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 11 \cdot b \cdot \overline{ba} = 2024.} Rezultă în continuare că 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 \cdot \overline{ba} = 184,} egalitate adevărată doar 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 b = 4} şi 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 = 6.} Aşadar, 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} = 64.}

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}}

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 '''Soluție:'''
-Egalitatea din enunţ se mai scrie <math>\overline{ab} - \overline{ba} = (a - b) \cdot (\overline{bb} \cdot \overline{ba} - 2015).</math> Descompunând <math>\overline{ab}</math> şi <math>\overline{ba}</math> în sistemul zecimal şi efectuând câteva calcule, obţinem:<math>9(a - b) = (a - b) \cdot (\overline{bb} \cdot \overline{ba} - 2015).</math>
+Egalitatea din enunţ se mai scrie <math>\overline{ab} - \overline{ba} = (a - b) \cdot (\overline{bb} \cdot \overline{ba} - 2015).</math> Descompunând <math>\overline{ab}</math> şi <math>\overline{ba}</math> în sistemul zecimal şi efectuând câteva calcule, obţinem: <math>9(a - b) = (a - b) \cdot (\overline{bb} \cdot \overline{ba} - 2015).</math>
 Cum <math>a - b \ne 0,</math> egalitatea devine <math>\overline{bb} \cdot \overline{ba} - 2015 = 9,</math> adică <math>11 \cdot b \cdot \overline{ba} = 2024.</math> Rezultă în continuare că <math>b \cdot \overline{ba} = 184,</math> egalitate adevărată doar pentru <math>b = 4</math> şi <math>a = 6.</math> Aşadar, <math>\overline{ab} = 64.</math>