Let $F$ be the set of all $n$-tuples $(A_{1}, A_{2}, \ldots, A_{n})$ where each $A_{i}$, $i=1,2, \ldots, n$ is a subset of $\{1,2, \ldots, 1998\}$. Let $|A|$ denote the number of elements of the set $A$. Find the number
$$
\sum_{(A_{1}, A_{2}, \ldots, A_{n})} |A_{1} \cup A_{2} \cup \ldots \cup A_{n}|.
$$
Let $ABC'$ be a triangle and $D$ the foot of the altitude from $A$. Let $E$ and $F$ be on a line passing through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let $M$ and $N$ be the midpoints of the line segments $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.
15 positive integers smaller than $1998$ are relatively prime (no pair has a common factor larger than $1$). Show that at least one of them must be prime.
Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go forwards and backwards along Martin Luther street until they arrive at their destination. What is the smallest value for a positive integer $K$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $KN$ blocks (no matter what $N$ turns out to be)? The mathematicians always walk together.
Two players play a game as follows. There are $n \gt 1$ rounds and $d \ge 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \ne m_1$. In round $k$ (for $k = 2, \dots, n$), A picks an integer $m_k$ such that $m_{k-1} \lt m_k \le m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} \lt n_k \le n_{k-1} + d$. A gets $\gcd(m_k, n_{k-1})$ points and B gets $\gcd(m_k, n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $n, d$ which player has a winning strategy?
Two players play a game as follows. The first player chooses two non-zero integers $A$ and $B$. The second player forms a quadratic with $A$, $B$ and $1998$ as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.
$ABC$ is a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.
Determine all positive integers $n$ for which there exists a set $S$ with the following properties:
(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;
(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.
Let $P$ be a polynomial of degree $6$ and let $a, b$ be real numbers such that $0 \lt a \lt b$. Suppose that $P(a) = P(-a)$, $P(b) = P(-b)$ and $P'(0) = 0$. Prove that $P(x) = P(-x)$ for all real $x$.
Is it possible to cover a $13 \times 13$ chessboard with forty-two tiles of size $4 \times 1$ so that only the central square of the chessboard remains uncovered? (It is assumed that each tile covers exactly four squares of the chessboard, and the tiles do not overlap.)
Let $n \geqslant 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.
We say that an integer $m$ covers the number 1998 if $1,9,9,8$ appear in this order as digits of $m$. (For instance, 1998 is covered by 215993698 but not by 213326798.) Let $k(n)$ be the number of positive integers that cover 1998 and have exactly $n$ digits ($n \geqslant 5$), all different from 0. What is the remainder of $k(n)$ in division by 8?
In a triangle $ABC$, $\angle BAC = 90^\circ$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA = 2 \angle BAD$. Prove that
$$
\frac{1}{|AD|} = \frac{1}{2} \left( \frac{1}{|BD|} + \frac{1}{|CD|} \right)
$$
In a convex pentagon $A B C D E$, the sides $A E$ and $B C$ are parallel and $\angle A D E=\angle B D C$. The diagonals $A C$ and $B E$ intersect at $P$. Prove that $\angle E A D=\angle B D P$ and $\angle C B D=\angle A D P$.
Let $n$ and $k$ be positive integers. There are $n k$ objects (of the same size) and $k$ boxes, each of which can hold $n$ objects. Each object is coloured in one of $k$ different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.
Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3, \ldots, 1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.