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Set de probleme cu nivel sporit de dificultate

 
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Claudiu Mindrila
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Joined: 01 Oct 2007
Posts: 557
Location: Targoviste

PostPosted: Tue Nov 25, 2008 10:58 pm    Post subject: Set de probleme cu nivel sporit de dificultate Reply with quote

1. Daca \frac{1}{a+2001}+\frac{1}{2b+2001}+\frac{1}{3c+2001}=\frac{3}{2002}, calculati \frac{a}{a+2001}+\frac{2b}{2b+2001}+\frac{3c}{3c+2001}.

2. Fie P={1,2,3,...,2003} si a,b,c,x,y \in P astfel incat:
\frac{a}{a+d}+\frac{b}{b+d}+\frac{c}{c+d}=x si \frac{d}{a+d}+\frac{b}{b+d}+\frac{d}{c+d}=y. Calculati x \cdot y
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Andi Brojbeanu
Newton


Joined: 22 Mar 2009
Posts: 383
Location: Targoviste (Dambovita)

PostPosted: Thu Apr 23, 2009 9:56 pm    Post subject: Reply with quote

1. \frac{a}{a+2001}+\frac{2b}{2b+2001}+\frac{3c}{3c+2001}=\frac{(a+2001)-2001}{a+2001}+\frac{(2b+2001)-2001}{2b+2001}+\frac{(3c+2001)-2001}{3c+2001}=1-\frac{2001}{a+2001}+1-\frac{2001}{2b+2001}+1-\frac{2001}{3c+2001}=3-2001(\frac{1}{a+2001}+\frac{1}{2b+2001}+\frac{1}{3c+2001})
=3-2001\cdot \frac{3}{2002}=\frac{6006-6003}{2002}=\frac{3}{2002}.

2. x+y=\frac{a}{a+d}+\frac{b}{b+d}+\frac{c}{c+d}+\frac{d}{a+d}+\frac{d}{b+d}+\frac{d}{c+d}=\frac{a+d}{a+d}+\frac{b+d}{b+d}+\frac{c+d}{c+d}=1+1+1=3.
Deoarece x,y \in \mathb{P}\rightarrow (x;y) \in \math {(1;2), (2;1)}\Rightarrow x\cdot y=1\cdot 2=2.
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