VMO 1991 — Division A

Determine all functions $f:\mathbb R\to\mathbb R$ such that, for arbitrary real numbers $x,y,z$,
$$\frac12 f(xy)+\frac12 f(xz)-f(x)f(yz)\ge\frac14.$$

Let an odd integer $k\gt 1$ be given. For each positive integer $n$, define $f(n)$ to be the largest nonnegative integer such that $k^n-1$ is divisible by $2^{f(n)}$. Determine a formula for $f(n)$ in terms of $k$ and $n$.

Let $Oxyz$ be a right trihedral angle, and let fixed points $A,B,C$ lie respectively on $Ox,Oy,Oz$. A variable sphere $(E)$ always passes through $A,B,C$ and meets the edges $Ox,Oy,Oz$ again at $A',B',C'$, respectively. Let $M,M'$ be the centroids of triangles $ABC,A'B'C'$, respectively. As $(E)$ varies, on what figure does the midpoint $S$ of $MM'$ move?

There are $1991$ students standing in a circle, facing inward, to play the following counting game. Each student says one number in turn clockwise, starting at a student $A$. The numbers said are $1,2,3$, repeated in that order. A student who says $2$ or $3$ must leave the circle immediately. The last remaining student receives a prize. Counting clockwise from $A$, who first says $1$, which starting position should a student choose to win?

Triangle $ABC$ has centroid $G$ and is inscribed in a circle of radius $R$. The medians from $A,B,C$, extended, meet the circle again at $D,E,F$, respectively. Prove that
$$\frac3R\le\frac1{GD}+\frac1{GE}+\frac1{GF}\le\sqrt3\left(\frac1{AB}+\frac1{BC}+\frac1{CA}\right).$$

Let positive numbers $x,y,z$ satisfy $x\ge y\ge z$. Prove that
$$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}\ge x^2+y^2+z^2.$$