Question bank VMO 1976

Find the integer solutions of
$$\begin{cases}x^{x+y}=y^{12},\\ y^{x+y}=x^3.\end{cases}$$

Determine the shape of triangle $ABC$ knowing that
$$\frac{a\cos A+b\cos B+c\cos C}{a\sin A+b\sin B+c\sin C}=\frac{a+b+c}{9R},$$
where $a,b,c$ are the lengths of $BC,CA,AB$ and $R$ is the circumradius of triangle $ABC$.

Prove that for every point $M$ inside triangle $ABC$,
$$d_ad_bd_c\le\frac{8S^3}{27abc},$$
where $d_a,d_b,d_c$ are the distances from $M$ to sides $BC,CA,AB$, respectively; $a,b,c$ are their lengths and $S$ is the area of the triangle. Extend this inequality to a tetrahedron in space.

Find a three-digit number equal to $1.5$ times the product of the factorials of its three digits (the factorial of $n$ means $1\cdot2\cdots n$).

Let $P$ be a plane and let $d,d'$ be two skew lines meeting $P$ at $A,A'$, respectively. Let $d''$ range over lines parallel to $P$ and meeting $d,d'$ at $M,M'$, respectively.

(a) Determine the position of $d''$ for which $MM'$ is shortest.

(b) Given a line $d''_0$, determine among the lines $d''$ the line $d''_1$ perpendicular to $d''_0$.

Let $k,n$ be positive integers and let $x_1,x_2,\ldots,x_k$ be positive numbers with sum $1$. Prove that
$$x_1^{-n}+x_2^{-n}+\cdots+x_k^{-n}\ge k^{n+1}.$$