VMO 1986

$a_1$, $a_2$, ... , $a_n$ are real numbers such that 1/2 ≤ $a_i$ ≤ 5 for each i. The real numbers $x_1$, $x_2$, ... , $x_n$ satisfy $4x_i^2$ - $4a_ix_i$ + ($a_i$ - 1)$^2$ = 0. Let S = ($x_1$ + $x_2$ + ... + $x_n$)/n, $S_2$ = ($x_1^2$ + $x_2^2$ + ... + $x_n^2$)/n. Show that √$S_2$ ≤ S + 1.

P is a pyramid whose base is a regular 1986-gon, and whose sloping sides are all equal. Its inradius is r and its circumradius is R. Show that R/r ≥ 1 + 1/cos(π/1986). Find the total area of the pyramid's faces when equality occurs.

The polynomial p(x) has degree n and p(1) = 2, p(2) = 4, p(3) = 8, ... , p(n+1) = $2^{n+1}$. Find p(n+2).

ABCD is a square. ABM is an equilateral triangle in the plane perpendicular to ABCD. E is the midpoint of AB. O is the midpoint of CM. The variable point X on the side AB is a distance x from B. P is the foot of the perpendicular from M to the line CX. Find the locus of P. Find the maximum and minimum values of XO.

Find all n > 1 such that ($x_1^2$ + $x_2^2$ + ... + $x_n^2$) ≥ $x_n$($x_1$ + $x_2$ + ... + $x_{n-1}$) for all real $x_i$.

A sequence of positive integers is constructed as follows. The first term is 1. Then we take the next two even numbers: 2, 4. Then we take the next three odd numbers: 5, 7, 9. Then we take the next four even numbers: 10, 12, 14, 16. And so on. Find the nth term of the sequence.