BG-MO 1990

Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?

Let be given a real number $\alpha\ne0$ . Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$ , segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).

Let $n=p_1p_2\cdots p_s$ , where $p_1,\ldots,p_s$ are distinct odd prime numbers. (a) Prove that the expression $$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$ where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients. (b) Prove that if $p$ is a prime divisor of $F_n(2)$ , then either $p\mid n$ or $n\mid p-1$ .

Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$ , one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$ .

Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$ . If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$ . When does equality hold?