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
Concursul "Nicolae Paun", 2008, problema 2
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Claudiu Mindrila
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Concursul "Nicolae Paun", 2008, problema 2
elev, clasa a X-a, C. N. "C-tin Carabella", Targoviste
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Marius Mainea
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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 \( x\in \mathbb{Q} \) \( x=\frac{1}{\sqrt{2009}-1}\cdot (\frac{x}{\sqrt{2}+1}+\frac{x}{\sqrt{3}+\sqrt{2}}+...+\frac{x}{\sqrt{2009}+\sqrt{2008}}) \)
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 \( x\in \mathbb{Q} \) \( x=\frac{1}{\sqrt{2009}-1}\cdot (\frac{x}{\sqrt{2}+1}+\frac{x}{\sqrt{3}+\sqrt{2}}+...+\frac{x}{\sqrt{2009}+\sqrt{2008}}) \)