Prove that the set $\{1,2,\ldots,1989\}$ can be expressed as the disjoint union of 17 subsets $A_1,A_2,\ldots,A_{17}$ such that:
(i) each $A_i$ contains the same number of elements;
(ii) the sum of all elements of each $A_i$ is the same for $i=1,2,\ldots,17$.
Let $ABC$ be a triangle. The bisector of angle $A$ meets the circumcircle of triangle $ABC$ in $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $AA_1$ meet the lines that bisect the two external angles at $B$ and $C$ in point $A^0$. Define $B^0$ and $C^0$ similarly. If $S_{X_1X_2\ldots X_n}$ denotes the area of the polygon $X_1X_2\ldots X_n$, prove that
$$S_{A^0B^0C^0}=2S_{AC_1BA_1CB_1}\ge4S_{ABC}.$$
Given a set $S$ in the plane containing $n$ points and satisfying the conditions
(i) no three points of $S$ are collinear,
(ii) for every point $P$ of $S$ there exist at least $k$ points in $S$ that have the same distance to $P$,
prove that the following inequality holds:
$$k\lt \frac12+\sqrt{2n}.$$
The quadrilateral $ABCD$ has the following properties:
(i) $AB=AD+BC$;
(ii) there is a point $P$ inside it at a distance $x$ from the side $CD$ such that $AP=x+AD$ and $BP=x+BC$.
Show that
$$\frac1{\sqrt x}\ge\frac1{\sqrt{AD}}+\frac1{\sqrt{BC}}.$$
For which positive integers $n$ does there exist a positive integer $N$ such that none of the integers $1+N,2+N,\ldots,n+N$ is the power of a prime number?
We consider permutations $(x_1,\ldots,x_{2n})$ of the set $\{1,\ldots,2n\}$ such that $|x_i-x_{i+1}|=n$ for at least one $i\in\{1,\ldots,2n-1\}$. For every natural number $n$, find out whether permutations with this property are more or less numerous than the remaining permutations of $\{1,\ldots,2n\}$.