a. Can you arrange the numbers $0, 1, \ldots, 9$ on the circumference of a circle, so that the difference between every pair of adjacent numbers is $3$, $4$ or $5$? For example, we can arrange the numbers $0, 1, \ldots, 6$ thus: $0, 3, 6, 2, 5, 1, 4$.
a. The digits of a natural number are rearranged and the resultant number is added to the original number. Prove that the answer cannot be $99\ldots9$ (1999 nines).
b. The digits of a natural number are rearranged and the resultant number is added to the original number to give $10^{10}$. Prove that the original number was divisible by $10$.
In the acute-angled triangle $ABC$, $AH$ is the longest altitude ($H$ lies on $BC$), $M$ is the midpoint of $AC$, and $CD$ is an angle bisector (with $D$ on $AB$).
a. If $AH \leq BM$, prove that the angle $ABC \leq 60^\circ$.
b. If $AH = BM = CD$, prove that $ABC$ is equilateral.
Four lighthouses are arbitrarily placed in the plane. Each has a stationary lamp which illuminates an angle of $90$ degrees. Prove that the lamps can be rotated so that at least one lamp is visible from every point of the plane.
$ABCD$ is a unit square. One vertex of a rhombus lies on side $AB$, another on side $BC$, and a third on side $AD$. Find the area of the set of all possible locations for the fourth vertex of the rhombus.
499 white rooks and a black king are placed on a $1000 \times 1000$ chess board. The rook and king moves are the same as in ordinary chess, except that taking is not allowed and the king is allowed to remain in check. No matter what the initial situation and no matter how white moves, the black king can always:
a) get into check (after some finite number of moves);
b) move so that apart from some initial moves, it is always in check after its move;
c) move so that apart from some initial moves, it is always in check (even just after white has moved).
What is the maximum possible length of a sequence of natural numbers $x_{1}, x_{2}, x_{3}, \ldots$ such that $x_{i} \leq 1998$ for $i \geq 1$, and $x_{i} = |x_{i - 1} - x_{i - 2}|$ for $i \geq 3$.
A natural number $k$ has the property that if $k$ divides $n$, then the number obtained from $n$ by reversing the order of its digits is also divisible by $k$. Prove that $k$ is a divisor of $99$.