VMO 1987

Let $x_n$ = (n+1)π/3974. Find the sum of all cos(± $x_1$ ± $x_2$ ± ... ± $x_{1987}$).

The sequences $a_0$, $a_1$, $a_2$, ... and $b_0$, $b_1$, $b_2$, ... are defined as follows. $a_0$ = 365, $a_{n+1}$ = $a_n$($a_n^{1986}$ + 1) + 1622, $b_0$ = 16, $b_{n+1}$ = $b_n$($b_n^3$ + 1) - 1952. Show that there is no number in both sequences.

There are n > 2 lines in the plane, no two parallel. The lines are not all concurrent. Show that there is a point on just two lines.

$x_1$, $x_2$, ... , $x_n$ are positive reals with sum X and n > 1. h ≤ k are two positive integers. H = $2^h$ and K = $2^k$. Show that $x_1^K$/(X - $x_1$)$^{H-1}$ + $x_2^K$/(X - $x_2$)$^{H-1}$ + $x_3^K$/(X - $x_3$)$^{H-1}$ + ... + $x_n^K$/(X - $x_n$)$^{H-1}$ ≥ $X^{K-H+1}$/( (n-1)$^{2H-1}n^{K-H}$). When does equality hold?

The function f(x) is defined and differentiable on the non-negative reals. It satisfies | f(x) | ≤ 5, f(x) f '(x) ≥ sin x for all x. Show that it tends to a limit as x tends to infinity.

Given 5 rays in space from the same point, show that we can always find two with an angle between them of at most $90^o$.