Azi am participat la barajul IMAC (Concurs ARHIMEDE). Sunt nesigur pe rezolvarile de la doua exercitii.

III.)
Pe un cerc se aleg punctele A, B si P. Din mijlocul M al arcului APB (nu ABP, nota mea!) este coborata perpendiculara MK pe cea mai mare dintre coardele AP si BP. Sa se demonstreze ca punctul K injumatateste linia franta APB.

II.) Iar al doilea arata cam asa:

*** QuickLaTeX cannot compile formula:

	
	\[\begin{array}{l}
	 {\rm{Sa se determine numerele reale pozitive }}{x_1},{x_2},{x_3},...,{x_n},{\rm{ }}n \ge 2{\rm{ pentru care}} \\
	 \left\{ \begin{array}{l}
	 {x_1} + {x_2} + {x_3} + ... + {x_n} = \frac{3}{2} \\
	 \frac{1}{{{x_1} + {x_2}}} + \frac{1}{{{x_2} + {x_3}}} + ... + \frac{1}{{{x_{n - 1}} + {x_n}}} + \frac{1}{{{x_n} + {x_1}}} = 3 \\
	 \end{array} \right. \\
	  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  \\
	 {\rm{Am mers pe varianta: 2}}\left( {{x_1} + {x_2} + {x_3} + ... + {x_n}} \right) = 2*\frac{3}{2} = 3 \Rightarrow 2{x_1} + 2{x_2} + 2{x_3} + ... + 2{x_{n - 1}} + 2{x_n} = \frac{1}{{{x_1} + {x_2}}} + \frac{1}{{{x_2} + {x_3}}} + ... + \frac{1}{{{x_{n - 1}} + {x_n}}} + \frac{1}{{{x_n} + {x_1}}} \\
	  \Rightarrow 2{x_1} + 2{x_2} + 2{x_3} + ... + 2{x_{n - 1}} + 2{x_n} - \frac{1}{{{x_1} + {x_2}}} - \frac{1}{{{x_2} + {x_3}}} - ... - \frac{1}{{{x_{n - 1}} + {x_n}}} = \frac{1}{{{x_n} + {x_1}}} \Rightarrow {x_1} + \underbrace {{x_1} + {x_2} - \frac{1}{{{x_1} + {x_2}}}}_{} + \underbrace {{x_2} + {x_3} - \frac{1}{{{x_2} + {x_3}}}}_{} + ... + \underbrace {{x_{n - 1}} + {x_n} - \frac{1}{{{x_{n - 1}} + {x_n}}}}_{} + {x_n} = \frac{1}{{{x_n} + {x_1}}} \\
	  \Rightarrow \underbrace {{x_1} + {x_2} - \frac{1}{{{x_1} + {x_2}}}}_{} + \underbrace {{x_2} + {x_3} - \frac{1}{{{x_2} + {x_3}}}}_{} + ... + \underbrace {{x_{n - 1}} + {x_n} - \frac{1}{{{x_{n - 1}} + {x_n}}}}_{} + \underbrace {{x_1} + {x_n} - \frac{1}{{{x_n} + {x_1}}}}_{} = 0 \\
	
	

*** Error message:
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Emergency stop.

Multumesc anticipat!