Page 1 of 1
Proprietati ale unei multimi finite
Posted: Tue Nov 25, 2008 10:54 pm
by Marius Mainea
Fie \( M\subset(0,\infty) \) o multime finita cu proprietatea ca orice element \( a\in M \) se poate reprezenta sub forma \( a=1+\frac{b}{c} \) cu \( b\in M \) si \( c\in M \).
Aratati ca exista \( x\in M \) si \( y\in M \) astfel incat \( x+y\ge 4 \)
Concursul ,,Nicolae Paun'' , 2004
Posted: Tue Nov 25, 2008 11:27 pm
by Claudiu Mindrila
Daca \( x \in M \Longrightarrow x=1+\frac{m}{n} \) cu \( x,y \in M \)
Putem alege \( y= 1+\frac{n}{m} \in M \) si vom obtine ca:
\( x+y=2+\frac{m}{n}+\frac{n}{m}\geq 2+2=4 \)
Posted: Tue Nov 25, 2008 11:31 pm
by Marius Mainea
Claudiu Mindrila wrote:
Putem alege \( y= 1+\frac{n}{m} \in M \)
De unde vine aceasta relatie?
Posted: Tue Dec 02, 2008 9:43 pm
by Marius Mainea
Fie m, minimul lui M.
Atunci \( m=1+\frac{b}{c}\ge 1+\frac{m}{c} \) si de aici
\( m\ge \frac{c}{c-1} \)
Asadar \( m+c\ge\frac{c}{c-1}+c\ge 4 \) si problema este rezolvata.