VMO 1986 — Division A

Consider $n$ inequalities
$$4x^2-4a_ix+(a_i-1)^2\le0,$$
where $a_i\in[\frac12,5]$ for $i=1,2,\ldots,n$. Let $x_i$ be any solution of the inequality with parameter $a_i$. Prove that
$$\frac{\sum_{k=1}^n x_k^2}{n}\le\frac{\sum_{k=1}^n x_k}{n}+1.$$

Let $R,r$ be the radii of the circumscribed and inscribed spheres of a regular $1986$-gonal pyramid.
(a) Prove that $\displaystyle\frac Rr\ge1+\frac1{\cos(\pi/1986)}$.
(b) Find its lateral surface area if its base side is $a$ and equality holds.

Let $M(y)$ be a polynomial of degree $n$ such that $M(y)=2^y$ for $y=1,2,\ldots,n+1$. Determine $M(n+2)$.

Let $ABCD$ be a square of side $2a$. In the plane through $AB$ perpendicular to the square's plane, construct an equilateral triangle $AMB$. A point $S$ moves along $AB$ with $SB=x$. Let $P$ be the orthogonal projection of $M$ onto the line $SC$, and let $E,O$ be the midpoints of $AB,CM$, respectively.
(a) Determine the locus of $P$ as $S$ moves along $AB$.
(b) Find the maximum and minimum values of $SO$.

Find all natural numbers $n\gt 1$ for which
$$\sum_{k=1}^n x_k^2\ge x_n\sum_{k=1}^{n-1}x_k$$
holds for every choice of $x_k$, $k=1,\ldots,n$.

From the natural numbers $1,2,3,\ldots$ form a sequence as follows: the first term is the odd number $1$; the next two terms are the even numbers $2,4$; the next three are the odd numbers $5,7,9$; the next four are the even numbers $10,12,14,16$; the next five are the odd numbers $17,19,21,23,25$; and so on. Determine the general term.