VMO 1980

Let $x_1$, $x_2$, ... , $x_n$ be real numbers in the interval [0, π] such that (1 + cos $x_1$) + (1 + cos $x_2$) + ... + (1 + cos $x_n$) is an odd integer. Show that sin $x_1$ + sin $x_2$ + ... + sin $x_n$ ≥ 1.

Let $x_1$, $x_2$, ... , $x_n$ be positive reals with sum s. Show that ($x_1$ + $1/x_1$)$^2$ + ($x_2$ + $1/x_2$)$^2$ + ... + ($x_n$ + $1/x_n$)$^2$ ≥ n(n/s + s/n)$^2$.

P is a point inside the triangle $A_1A_2A_3$. The ray $A_i$P meets the opposite side at $B_i$. $C_i$ is the midpoint of $A_iB_i$ and $D_i$ is the midpoint of $PB_i$. Show that area $C_1C_2C_3$ = area $D_1D_2D_3$.

Show that for any tetrahedron it is possible to find two perpendicular planes such that if the projection of the tetrahedron onto the two planes has areas A and A', then A'/A > √2.

Does there exist real m such that the equation $x^3$ - $2x^2$ - 2x + m has three different rational roots?

Given n > 1 and real s > 0, find the maximum of $x_1x_2$ + $x_2x_3$ + $x_3x_4$ + ... + $x_{n-1}x_n$ for non-negative reals $x_i$ such that $x_1$ + $x_2$ + ... + $x_n$ = s.