Školjka

%V0 Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ $$a \cos^2{x}+b \cos{x}+c=0.$$ Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

%V0 Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

%V0 An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$; b) Calculate the distance of $p$ from either base; c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

%V0 Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

%V0 In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $$a_{1}\geq a_{2}\geq \dots \geq a_{n}.$$ Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

%V0 Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.