VMO 1964

Given an arbitrary angle $\alpha$ , compute $cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ . Generalize this result and justify your answer.

Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$ . Find the number of roots of the equation $x + | x^2 - 1 | = k$ , where $k$ is a real constant.

Let $P$ be a plane and two points $A \in (P),O \notin (P)$ . For each line in $(P)$ through $A$ , let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$ . Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$ . Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$ , intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$ .

Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$ . Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$ , where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$ .