O

1. Calculati

(1)
\begin{align} a) \ N = \left 8 \over \ 7 \right + \left {8 \ . \ 6} \over \ 7 \ . \ 5 \right + \left 8 \ . \ 6 \ . \ 4 \over \ 7 \ . \ 5 \ . \ 3 \right + \left 8 \ . \ 6 \ . \ 4 \ . \ 2 \over \ 7 \ . \ 5 \ . \ 3 \ . \ 1 \right \end{align}
(2)
\begin{align} b) \ S = \left 2010 \over \ 2009 \right + \left {2010 \ . \ 2008} \over \ 2009 \ . \ 2007 \right + \left 2010 \ . \ 2008 \ . \ 2006 \over \ 2009 \ . \ 2007 \ . \ 2005 \right + \cdots + \left 2010 \ . \ 2008 \ . \cdots\ . \ 2 \over \ 2009 \ . \ 2007 \ . \cdots\ .\ 1 \right \.\ \end{align}

Raspuns_1

2. Aratati ca

(3)
\begin{align} \frac{1}{1^4 + 1^2 + 1} + \frac{2}{2^4 + 2^2 +1}+ \cdots \ + \frac{2010}{2010^4 +2010^2 +1}< \frac{1}{2} \end{align}

Indicatie_2

3. Se considera numerele naturale a1, a2, …, a102 astfel ca 0< a1< a2< …< a102<307.

a) Cate valori distincte poate lua suma a1+ a2 + …+ a102 ?

b) Efectuand toate diferentele a oricaror doua numere vecine ak+1 - ak, aratati ca una dintre acestea se repeta de cel putin 21 de ori.

Rezolvare

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