CZ-SK-MO 1994

Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$ . Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$ .

A cuboid of volume $V$ contains a convex polyhedron $M$ . The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$ ?

A convex $1994$ -gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have (a) $991$ diagonals of length $3$ and $6$ of length $2$ ? (b) $985$ diagonals of length $6, 4$ of length $8$ , and $8$ of length $3$ ?

Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$ , the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$ . Prove that for any finite set $P$ of prime numbers the following inequality holds: $$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$

In an acute-angled triangle $ABC$ , the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$ . If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that $$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab}$$