VMO 1998

Define the sequence $x_1$, $x_2$, $x_3$, ... by $x_1$ = a ≥ 1, $x_{n+1}$ = 1 + ln($x_n$($x_n^2$+3)/(1 + $3x_n^2$) ). Show that the sequence converges and find the limit.

Let O be the circumcenter of the tetrahedron ABCD. Let A', B', C', D' be points on the circumsphere such that AA', BB', CC' and DD' are diameters. Let A" be the centroid of the triangle BCD. Define B", C", D" similarly. Show that the lines A'A", B'B", C'C", D'D" are concurrent. Suppose they meet at X. Show that the line through X and the midpoint of AB is perpendicular to CD.

The sequence $a_0$, $a_1$, $a_2$, ... is defined by $a_0$= 20, $a_1$ = 100, $a_{n+2}$ = $4a_{n+1}$ + $5a_n$ + 20. Find the smallest m such that $a_m$ - $a_0$, $a_{m+1}$ - $a_1$, $a_{m+2}$ - $a_2$, ... are all divisible by 1998.

Does there exist an infinite real sequence $x_1$, $x_2$, $x_3$, ... such that | $x_n$ | ≤ 0.666, and | $x_m$ - $x_n$ | ≥ 1/($n^2$ + n + $m^2$ + m) for all distinct m, n?

What is the minimum value of √( (x+1)$^2$ + (y-1)$^2$) + √( (x-1)$^2$ + (y+1)$^2$) + √( (x+2)$^2$ + (y+2)$^2$)?

Find all positive integers n for which there is a polynomial p(x) with real coefficients such that p($x^{1998}$ - $x^{-1998}$) = ($x^n$ - $x^{-n}$) for all x.