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Concursul "Nicolae Paun", 2008, problema 2
Posted: Sat Dec 13, 2008 8:06 pm
by Claudiu Mindrila
a) Sa se arate ca orice numar real se poate scrie ca produs de doua numere irationale diferite si ca un produs de trei numere irationale diferite.
b) Sa se arate ca orice numar real se poate scrie ca suma de \( 2008 \) numere rationale diferite.
Lavinia Savu
Posted: Tue Dec 16, 2008 1:29 pm
by Marius Mainea
In primul rand trebuia nr real nenul
a) daca \( x\in \mathbb{Q} \)
\( x=\frac{x}{\sqrt{2}}\cdot \sqrt{2} \) , \( x=\frac{x}{\sqrt{3}}\cdot \sqrt{3} \) \( x=\frac{x}{\sqrt[3]{6}}\cdot \sqrt[3]{2}\cdot \sqrt[3]{3} \)
Daca \( x\notin\mathbb{Q} \) \( x=sgn(x)\cdot \sqrt{\frac{|x|}{2}}\cdot \sqrt{2|x|} \)
\( x=\sqrt[3]{\frac{x}{6}}\cdot \sqrt[3]{2x}\cdot \sqrt[3]{3x} \)
In al doilea rand la b) trebuia nr irationale diferite.
b) Daca \( x\notin\mathbb{Q} \) \( s=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{2008} \)
atunci \( x=\frac{x}{s}+\frac{x}{2s}+\frac{x}{3s}+....+\frac{x}{2008s} \)
Daca \( x\in \mathbb{Q} \) \( x=\frac{1}{\sqrt{2009}-1}\cdot (\frac{x}{\sqrt{2}+1}+\frac{x}{\sqrt{3}+\sqrt{2}}+...+\frac{x}{\sqrt{2009}+\sqrt{2008}}) \)