Let $\alpha$ be an arbitrary angle. Determine the value of the following expression:
$$\cos\alpha+\cos\left(\alpha+\frac{2\pi}{3}\right)+\cos\left(\alpha+\frac{4\pi}{3}\right).$$
Draw the graphs of the following functions:
$$u=\sqrt{(x^2-1)^2},\qquad y=x+\sqrt{(x^2-1)^2},\qquad x\in\mathbb R.$$
Discuss the number of solutions of
$$x+\sqrt{(x^2-1)^2}=m$$
as a function of the parameter $m$.
From a point $O$ outside a plane $(P)$, perpendiculars $OH$ are dropped to the lines in $P$ passing through a fixed point $A$. Determine the locus $(c)$ of $H$.
Let $(C)$ be the oblique cone with vertex $O$ and base $(c)$. Prove that planes parallel to $(P)$ or perpendicular to $OA$ cut $(C)$ in circles.
$(C)$ has two planes of symmetry that cut it in angles $\alpha,\beta$. Determine the relation between $\alpha$ and $\beta$.
Editorial conventions: proper circles and a nondegenerate cone require $A$ to differ from the perpendicular projection of $O$ onto $P$, and the section planes must not pass through $O$. “Angles” means full apex angles in the two axial symmetry planes. We state the relation with the smaller one labeled $\alpha$; the source does not specify their order.
A tree branches as follows: after growing for one year it begins to branch; thereafter, every two years it produces another branch. This rule also applies to every new branch: one year after it appears it produces a child branch, and that main branch then produces another branch every two years. Count the main trunk as a special branch. Find the number of branches during the fifth year.
Let $S_n$ be the number of branches during year $n$. Prove
$$S_n=S_{n-1}+S_{n-2}.$$
Let $a\gt b$ satisfy $a+b=1$, $ab=-1$. Prove by induction that
$$S_n=\frac1{\sqrt5}(a^{n+1}-b^{n+1}).$$
Indexing clarification: as required by the displayed formula, the first year contains just the trunk ($S_1=1$); the first new branch appears at its end, so $S_2=2$.