Cristian Calude, proba pe echipe, R.IV, P.II

[Laurian Filip]

Sa se determine multimile nevide \( A \subset \mathbb{R}^* \) care satisfac proprietatile:
i) A are cel mult 5 elemente
ii) \( x \in A \) implica \( \frac{1}{x} \in A \) si \( 1-x \in A \)

[alex2008]

Ciudat , parca era la clasa a IX-a .
\( x\in A \Rightarrow \frac{1}{x} \in A \) si \( 1-x \in A \)
\( \frac{1}{x}\in A \Rightarrow \frac{1}{\frac{1}{x}} \) si \( 1-\frac{1}{x} \Rightarrow x\in A \) si \( \frac{x-1}{x} \in A \)
\( \frac{x-1}{x}\in A \Rightarrow \frac{x}{x-1}\in A \) si \( 1-\frac{x-1}{x} \in A \Rightarrow \frac{1}{x} \in A \)
\( \frac{x}{x-1} \in A \Rightarrow \frac{x-1}{x} \in A \) si \( 1-\frac{x}{x-1} \in A \Rightarrow \frac{1}{1-x} \in A \)
\( \frac{1}{1-x} \in A \Rightarrow 1-x \in A \) si \( 1-\frac{1}{1-x} \in A \Rightarrow \frac{x}{x-1}\in A \)
\( \Rightarrow A=\{x , 1-x , \frac{1}{x} , \frac{x-1}{x} , \frac{x}{x-1} , \frac{1}{1-x}\} \)
A are cel mult 5 elemente
\( \Rightarrow \) Cazul I. \( x=1-x \Rightarrow x=\frac{1}{2} \Rightarrow 2\in A , -1\in A
\Rightarrow A=\{-1,\frac{1}{2},2\} \)
Cazul II. \( x=\frac{1}{x} \Rightarrow x=\pm1 \Rightarrow 0\in A (F) \)
Cazul III. \( x=\frac{x-1}{x} \Rightarrow x^2-x+1=0 \Rightarrow \Delta\le0 \)
Cazul IV. \( x=\frac{x}{x-1} \Rightarrow x=0 \) sau \( x=2 \)
Cazul V. \( x=\frac{1}{1-x} \Rightarrow x^2-x+1=0 \)
Cazul VI. \( 1-x=\frac{1}{x} \Rightarrow x^2-x+1=0 \)
Cazul VII. \( 1-x=\frac{x-1}{x} \Rightarrow x=\pm1 \)
Cazul VIII. \( 1-x=\frac{x}{x-1} \Rightarrow x^2-x+1=0 \)
Cazul IX. \( 1-x=\frac{1}{1-x} \Rightarrow x=\frac{1}{2} \)
Cazul X. \( \frac{1}{x}=\frac{x-1}{x} \Rightarrow x=0 \) sau \( x=2 \)
Cazul XI. \( \frac{1}{x}=\frac{x}{x-1} \Rightarrow x^2-x+1=0 \)
Cazul XII. \( \frac{1}{x}=\frac{1}{1-x} \Rightarrow x=\frac{1}{2} \)
Cazul XIII. \( \frac{x-1}{x}=\frac{x}{x-1} \Rightarrow x=\frac{1}{2} \)
Cazul XIV. \( \frac{x-1}{x}=\frac{1}{1-x} \Rightarrow x^2-x+1=0 \)
Cazul XV. \( \frac{x}{x-1}=\frac{1}{1-x} \Rightarrow x=\pm1 \)