AT-OEMO-REG 2004

Determine all integers $a$ and $b$ , so that $(a^3+b)(a+b^3)=(a+b)^4$

Solve the following equation for real numbers: $\sqrt{4-x\sqrt{4-(x-2)\sqrt{1+(x-5)(x-7)}}}=\frac{5x-6-x^2}{2}$ (all square roots are non negative)

Given is a convex quadrilateral $ABCD$ with $\angle ADC=\angle BCD>90^{\circ}$ . Let $E$ be the point of intersection of the line $AC$ with the parallel line to $AD$ through $B$ and $F$ be the point of intersection of the line $BD$ with the parallel line to $BC$ through $A$ . Show that $EF$ is parallel to $CD$

The sequence $< x_n >$ is defined through: $x_{n + 1} = \left(\frac {n}{2004} + \frac {1}{n}\right)x_n^2 - \frac {n^3}{2004} + 1$ for $n > 0$ Let $x_1$ be a non-negative integer smaller than $204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.