BALTIC-WAY 1992

Let $p$ and $q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least three positive integers greater than 1 (not necessarily different).

Denote by $d(n)$ the number of all positive divisors of a positive integer $n$ (including 1 and $n$ ). Prove that there are infinitely many $n$ such that $\frac{n}{d(n)}$ is an integer.

Find an infinite non-constant arithmetic progression of positive integers such that each term is neither a sum of two squares, nor a sum of two cubes (of positive integers).

Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?

Given that $a^{2}+b^{2}+(a+b)^{2}=c^{2}+d^{2}+(c+d)^{2}$, prove that $a^{4}+b^{4}+(a+b)^{4}=c^{4}+d^{4}+(c+d)^{4}$.

Prove that the product of the 99 numbers of the form $\frac{k^{3}-1}{k^{3}+1}$ where $k=2,3, \ldots, 100$, is greater than $\frac{2}{3}$.

Let $a=\sqrt[1992]{1992}$. Which number is greater:

$$\left.a^{a^{a^{a}}}\right\} 1992$$

or 1992 ?

Find all integers satisfying the equation $2^{x} \cdot(4-x)=2 x+4$.

A polynomial $f(x)=x^{3}+a x^{2}+b x+c$ is such that $b<0$ and $a b=9 c$. Prove that the polynomial has three different real roots.

Find all fourth degree polynomials $p(x)$ such that the following four conditions are satisfied:

(i) $p(x)=p(-x)$ for all $x$.

(ii) $p(x) \geq 0$ for all $x$.

(iii) $p(0)=1$.

(iv) $p(x)$ has exactly two local minimum points $x_{1}$ and $x_{2}$ such that $\left|x_{1}-x_{2}\right|=2$.

Let $\mathbb{Q}^{+}$denote the set of positive rational numbers. Show that there exists one and only one function $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$satisfying the following conditions:

(i) If $0<q<\frac{1}{2}$ then $f(q)=1+f\left(\frac{q}{1-2 q}\right)$.

(ii) If $1<q \leq 2$ then $f(q)=1+f(q-1)$.

(iii) $f(q) \cdot f\left(\frac{1}{q}\right)=1$ for all $q \in \mathbb{Q}^{+}$.

Let $\mathbb{N}$ denote the set of positive integers. Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be a bijective function and assume that there exists a finite limit

$$\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n}=L$$

What are the possible values of $L$ ?

Prove that for any positive $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$ the inequality

$$\sum_{i=1}^{n} \frac{1}{x_{i} y_{i}} \geq \frac{4 n^{2}}{\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)^{2}}$$

holds.

There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from the other one. Prove that there is a town such that all the remaining towns can be reached from it.

Noah has to fit 8 species of animals into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accommodation. Prove that there is a way to assign the animals to their cages so that each species shares a cage with compatible species.

All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

Quadrangle $A B C D$ is inscribed in a circle with radius 1 in such a way that one diagonal, $A C$, is a diameter of the circle, while the other diagonal, $B D$, is as long as $A B$. The diagonals intersect in $P$. It is known that the length of $P C$ is $\frac{2}{5}$. How long is the side $C D$ ?

Figure 1

Figure 2

Figure 3

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

Let $C$ be a circle in the plane. Let $C_{1}$ and $C_{2}$ be non-intersecting circles touching $C$ internally at points $A$ and $B$ respectively. Let $t$ be a common tangent of $C_{1}$ and $C_{2}$, touching them at points $D$ and $E$ respectively, such that both $C_{1}$ and $C_{2}$ are on the same side of $t$. Let $F$ be the point of intersection of $A D$ and $B E$. Show that $F$ lies on $C$.

Let $a \leq b \leq c$ be the sides of a right triangle, and let $2 p$ be its perimeter. Show that

$$p(p-c)=(p-a)(p-b)=S$$

where $S$ is the area of the triangle.