MD-MO 2007

Define the sequence $(x_{n})$ : $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$ . Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$ , wehere $[$ $]$ denotes the integer part.

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$ . Find $\lim_{n\rightarrow\infty}a_{n}$ .

$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$ . Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$ . $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$ , respectively. Find the minimum value of the $CP+PQ$ , where $P\in[GH]$ and $Q\in[EF]$ .

The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$ , for any $x\in(0,\pi)$ . Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$ , $g(x)=f(x)\cdot f(1-x)$ .

Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$ , for all $i=\overline{1,n}$ . Prove the inequality: $$\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).$$

Define $(b_{n})$ to be: $b_{0}=12$ , $b_{1}=\frac{\sqrt{3}}{2}$ adn $b_{n+1}+b_{n-1}=b_{n}\cdot\sqrt{3}$ . Find $b_{0}+b_{1}+\dots+b_{2007}$ .

Note. Maybe this seems too easy, but I want to post all the problems...

Given a tetrahedron $VABC$ with edges $VA$ , $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$ . Find the maximal volume of $VABC$ .