E:16610

E:16610 (Cristina Vijdeluc, Salonic și Mihai Vijdeluc)

Aflați 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{abc}} , știind 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 \overline{ab}} 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 \overline{cb}} sunt numere prime 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 \overline{ab}^2 + \overline{cb}^2 + b^2 = 2027} .

Soluție

Din relația dată deducem 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 \overline{ab} \leq 43} , 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{cb} \leq 43} . În plus, cum acestea sunt numere prime de două cifre, trebuie să fie impare nedivizibile cu 5, prin urmare $b\in \{1,3,7,9\}$ . Ținând cont de ultima cifră, rămâne $b=3$ sau $b=7$ . Pentru $b=7$ avem ${\overline {ab}},{\overline {cb}}\in \{17,37\}$ , care nu verifică. Pentru $b=3$ avem ${\overline {ab}},{\overline {cb}}\in \{13,23,43\}$ , dintre care verifică ${\overline {ab}}=13$ , ${\overline {cb}}=43$ și ${\overline {ab}}=43$ , ${\overline {cb}}=13$ .

Prin urmare, ${\overline {abc}}$ 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 134} 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 431} .