Question bank VMO 1979

Prove that for every $x\gt 1$ there exists a triangle whose side lengths are $P_1(x)=x^4+x^3+2x^2+x+1$, $P_2(x)=2x^3+x^2+2x+1$, and $P_3(x)=x^4-1$. Prove that the largest angles of all these triangles are equal, and determine their common value.

The equation $x^3+ax^2+bx+c=0$ has three real roots $t,u,v$, not necessarily distinct. For which values of $a,b,c$ do the numbers $t^3,u^3,v^3$ satisfy $x^3+a^3x^2+b^3x+c^3=0$?

Divide a triangular garden $ABC$, whose three sides have different lengths, by a segment $AM$ so that the ratio of the areas of triangles $ABM$ and $ACM$ equals the ratio of their corresponding perimeters.

Prove that for every positive integer $n$, $2\cos n\theta$ is a degree-$n$ polynomial in $2\cos\theta$.

Find all real numbers $\alpha$ for which $x^2-2x[x]+x-\alpha=0$ has two nonnegative roots, where $[x]$ denotes the integer part (floor) of $x$.

In space, two congruent rectangles $ABCD$ and $ABEF$ have $AB=m$ and $BC=BE=m\sqrt2$. Determine the relative position of their planes for which $CA$ is perpendicular to $BF$. In that case, determine the position and dimensions of a regular tetrahedron with two vertices on $CA$ and its other two vertices on $BF$.