VWO 2006

(a) Solve for $\theta\in\mathbb{R}$ : $\cos(4\theta) = \cos(3\theta)$

(b) $\cos\left(\frac{2\pi}{7}\right)$ , $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$ .

Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$ . $Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$ . $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$ . And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$ . $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$ . $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$ . And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$ . Determine $\frac{|PB|}{|AB|}$ if $S=S'$ .

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Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?

Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that

$$f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}.$$