Školjka

%V0 In the plane we are given a set $E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $E,$ there exist at least 1593 other points of $E$ to which it is joined by a path. Show that there exist six points of $E$ every pair of which are joined by a path. Alternative version: Is it possible to find a set $E$ of 1991 points in the plane and paths joining certain pairs of the points in $E$ such that every point of $E$ is joined with a path to at least 1592 other points of $E,$ and in every subset of six points of $E$ there exist at least two points that are not joined?

%V0 Find the highest degree $k$ of $1991$ for which $1991^k$ divides the number $$1990^{1991^{1992}} + 1992^{1991^{1990}}.$$

%V0 Let $\alpha$ be a rational number with $0 < \alpha < 1$ and $\cos (3 \pi \alpha) + 2\cos(2 \pi \alpha) = 0$. Prove that $\alpha = \frac {2}{3}$.

%V0 Let $f(x)$ be a monic polynomial of degree $1991$ with integer coefficients. Define $g(x) = f^2(x) - 9.$ Show that the number of distinct integer solutions of $g(x) = 0$ cannot exceed $1995.$

%V0 Real constants $a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $y = x^3 + ax^2 + bx + c.$ Prove that the square has sides of length $\sqrt[4]{72}.$

%V0 Determine the maximum value of the sum $$\sum_{i < j} x_ix_j (x_i + x_j)$$ over all $n -$tuples $(x_1, \ldots, x_n),$ satisfying $x_i \geq 0$ and $\sum^n_{i = 1} x_i = 1.$