E:16380

E:16380 (Cristina Vijdeluc, Salonic şi Mihai Vijdeluc, Baia Mare)

Aflaţi 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 a,b,c,d} pentru care are loc relaţia 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 2(3^{a + 1} + 3^{b + 1} + 3^{c + 1}) = 3 \cdot 6 \cdot 9 \cdot \ldots \cdot d.}

Soluție:

Egalitatea din enunţul se poate scrie 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 6(3^a + 3^b +3^c) = 3 \cdot 6 \cdot 9 \cdot \ldots \cdot d.} Trebuie ca 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 \geqslant 6} şi $d$ să fie divizibil cu $3.$

Dacă $d=6,$ atunci $3^{a}+3^{b}+3^{c}=3,$ ceea ce înseamnă $a=b=c=0.$

Dacă $d=9,$ atunci $3^{a}+3^{b}+3^{c}=27,$ de unde rezultă că $a=b=c=2.$

Dacă $d=12,$ obţinem $3^{a}+3^{b}+3^{c}=3^{4}\cdot 4,$ adică $3^{a-4}+3^{b-4}+3^{c-4}=4,$ ecuaţia care nu are soluţii.

Pentru $d\geqslant 15,$ ultima cifră a produsului din membrul drept este zero. Dar 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 u(3^n) \in \{1,3,7,9\},} deci o sumă de trei puteri ale lui 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 3} nu are niciodată ultima cifră zero.

Aşadar, soluţiile căutate sunt 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, c, d) \in \{(0, 0, 0, 6),(2, 2, 2, 9)\}.}