DE-MO 2009

Find all non-negative real numbers $a$ such that the equation $$\sqrt[3]{1+x}+\sqrt[3]{1-x}=a$$ has at least one real solution $x$ with $0 \leq x \leq 1$ . For all such $a$ , what is $x$ ?

Find all positive interger $n$ so that $n^3-5n^2+9n-6$ is perfect square number.

Let $ABCD$ be a (non-degenerate) quadrangle and $N$ the intersection of $AC$ and $BD$ . Denote by $a,b,c,d$ the length of the altitudes from $N$ to $AB,BC,CD,DA$ , respectively.

Prove that $\frac{1}{a}+\frac{1}{c} = \frac{1}{b}+\frac{1}{d}$ if $ABCD$ has an incircle.

Extension: Prove that the converse is true, too.

[If this has already been posted, I humbly apologize. A quick search turned up nothing.]

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$ , such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

Let a triangle $ABC$ . $E,F$ in segment $AB$ so that $E$ lie between $AF$ and half of circle with diameter $EF$ is tangent with $BC,CA$ at $G,H$ . $HF$ cut $GE$ at $S$ , $HE$ cut $FG$ at $T$ . Prove that $C$ is midpoint of $ST$ .

Let a sequences: $x_0\in [0;1],x_{n+1}=\frac56-\frac43 \Big|x_n-\frac12\Big|$ . Find the "best" $|a;b|$ so that for all $x_0$ we have $x_{2009}\in [a;b]$