VMO 1989

Show that the absolute value of sin(kx) /N + sin(kx + x) /(N+1) + sin(kx + 2x) /(N+2) + ... + sin(kx + nx) /(N+n) does not exceed the smaller of (n+1) |x| and 1/(N sin(x/2) ), where N is a positive integer and k is real and satisfies 0 ≤ k ≤ N.

Let $a_1$ = 1, $a_2$ = 1, $a_{n+2}$ = $a_{n+1}$ + $a_n$ be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that $1985a_n^2$ + $1956a_n$ + 1960 is divisible by 1989. Does there exist a term such that $1985a_n^2$ + $1956a_n$ + 1960 + 2 is divisible by 1989?

ABCD is a square side 2. The segment AB is moved continuously until it coincides with CD (note that A is brought into coincidence with the opposite corner). Show that this can be done in such a way that the region passed over by AB during the motion has area < 5π/6.

Do there exist integers m, n not both divisible by 5 such that $m^2$ + $19n^2$ = 198·$10^{1989}$?

Define the sequence of polynomials $p_0$(x), $p_1$(x), $p_2$(x), ... by $p_0$(x) = 0, $p_{n+1}$(x) = $p_n$(x) + (x - $p_n$(x)$^2$)/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - $p_n$(x) ≤ 2/(n+1).

ABCDA'B'C'D' is a parallelepiped (with edges AB, BC, CD, DA, AA', BB', CC', DD', A'B', B'C', C'D', D'A'). Show that if a line intersects three of the lines AB', BC', CA', AD', then it also intersects the fourth.