36th Vietnam 1998 problems

A1.  Define the sequence x1, x2, x3, ... by x1 = a ≥ 1, xn+1 = 1 + ln(xn(xn2+3)/(1 + 3xn2) ). Show that the sequence converges and find the limit.
A2.  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.
A3.  The sequence a0, a1, a2, ... is defined by a0= 20, a1 = 100, an+2 = 4an+1 + 5an + 20. Find the smallest m such that am - a0, am+1 - a1, am+2 - a2, ... are all divisible by 1998.
B1.  Does there exist an infinite real sequence x1, x2, x3, ... such that | xn | ≤ 0.666, and | xm - xn | ≥ 1/(n2 + n + m2 + m) for all distinct m, n?
B2.  What is the minimum value of √( (x+1)2 + (y-1)2) + √( (x-1)2 + (y+1)2) + √( (x+2)2 + (y+2)2)?
B3.  Find all positive integers n for which there is a polynomial p(x) with real coefficients such that p(x1998 - x-1998) = (xn - x-n) for all x.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
22 July 2002
Last corrected/updated 7 Mar 04