ÖMO 1998

Let $a \geq 0$ be a natural number. Determine all rational $x$ , so that $$\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}$$ All occurring square roots, are not negative.Note. It seems the set of natural numbers = $\mathbb N = \{0,1,2,\ldots\}$ in this problem.

Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$ . What is the highest power of $98$ , that a) $Q_n$ , b) $K_n$ or c) $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$ , then one get a number $N$ . By which power-of-two number is $N$ still divisible?

Let $a_n$ be a sequence recursively defined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$ . Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$ . For what value of the base $b$ we get the sum $1$ ?

Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$ , are there?

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$ .(a) Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$ .(b) What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$ . What is the relationship between the lengths of the diagonals $AC$ and $BD$ ?