Question bank VMO 1977

Solve the inequality
$$\sqrt{x-\frac1x}-\sqrt{1-\frac1x}\gt \frac{x-1}{x}.$$

Prove that there exist 1977 triangles whose angles $A,B,C$ satisfy
$$\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\frac{12}{7},\qquad \sin A\sin B\sin C=\frac{12}{25}.$$

Into how many regions do $n$ circles divide the plane if every pair meets at two distinct points and no three circles have a common point?

Find necessary and sufficient conditions for the function $f(t)=mt^3+nt^2+pt+q$ to have an integer value for every value of $t$.

Let $a_0,a_1,\ldots,a_n,a_{n+1}$ be a finite sequence of real numbers satisfying $a_0=a_{n+1}=0$ and
$$|a_{k-1}-2a_k+a_{k+1}|\le1.$$
Prove that
$$a_k\le\frac{k(n+1-k)}2\qquad(k=0,1,\ldots,n+1).$$

An $m$-sided polygon $(P)$ and an $n$-sided polygon $(P')$ lie in two parallel planes. Consider the family of segments $AA'$ with $A$ and $A'$ lying on sides of $(P)$ and $(P')$, respectively. Explain how to find:
(a) a shortest segment $AA'$;
(b) a longest segment $AA'$.