Question

COROANĂ !


Determinați Nr. naturale nenule a,b,c (0 <a<d<c) și d€N, știind că a,b,c sunt direct proporționale cu 3,4 și d,iar 3a+4b+dc mai mic sau egal cu 50.


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Answer 1

$\displaystyle\\\left\{a,b,c\right\} \textnormal{d.p} \left\{3,4,d\right\} \Longleftrightarrow \frac{a}{3}=\frac{b}{4}=\frac{c}{d}=k,~k\in\mathbb{N^*}~(*).\\\stackrel{(*)}\Longrightarrow a=3k,~b=4k~\textnormal{iar}~c=dk,\textnormal{~evident,~}d\neq 0.\\3a+4b+dc\leq 50 \Longleftrightarrow (25+d^2)k\leq 50,~\textnormal{dar},~26\leq 25+d^2 |\cdot k \Longleftrightarrow\\26k\leq (25+d^2)k\leq 50 \Longleftrightarrow 26k \leq 50 \Longleftrightarrow k\leq 1,~\textnormal{cum}~k\in\mathbb{N} \Longrightarrow$

$\displaystyle\\k\in\left\{0,1\right\},~\textnormal{se verifica usor ca daca k=0 obtinem ca a=b=c=0}\\\textnormal{ceea ce conduce la o contradictie.}\\\textnormal{Asadar, ramane de verificat k=1}.\\k=1 \Longrightarrow d\in\left\{1,2,3,4,5\right\},~\textnormal{dar~cum~c=d, }0<a<b<c, \textnormal{~iar~a=3, b=4} \\\implies d=c=5,~\textnormal{deci,}~(a,b,c,d)\in\left\{(3,4,5,5 )\right\}.$