VMO 1982 — Division A

Construct a quadratic equation with integer coefficients whose two roots are $\cos72^\circ$ and $\cos144^\circ$.

Solve over the real numbers and discuss according to the real parameter $m$:
$$x(x+1)(x+2)(x+3)-m+1=0.$$

Let $ABC$ be a triangle. Construct an equilateral triangle externally on each of its three sides, and let $\Delta$ be the triangle whose vertices are the centers of these three equilateral triangles. Similarly construct equilateral triangles internally on the sides of $ABC$, and let $\Delta'$ be the triangle of their centers. Prove that the area of $ABC$ equals the difference between the areas of $\Delta$ and $\Delta'$.

Find the positive integer solutions of
$$2^x+2^y+2^z=2336,\qquad x\lt y\lt z.$$

Let $p$ be a positive integer and let $q,z$ be real numbers. Suppose $q^{p+1}\le z\le1$ and $0\le q\le1$. Prove that
$$\prod_{k=1}^p\left|\frac{z-q^k}{z+q^k}\right|\le\prod_{k=1}^p\frac{1-q^k}{1+q^k}.$$

Let $ABCD.A'B'C'D'$ be a cube. Prove that no line intersects all four of the following lines: $AA'$, $BC$, $D'C'$, and the line joining the midpoints of the edges $BB'$ and $DD'$.