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1. 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 ≤ BM, prove that the angle ABC ≤ 60.
(b) If AH = BM = CD, prove that ABC is equilateral.
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2. (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 ... 9 (1999 nines).
(b) The digits of a natural number are rearranged and the resultant number is added to the original number to give 1010. Prove that the original number was divisible by 10.
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3. Four lighthouses are arbitarily 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.
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4. (a) Can you arrange the numbers 0, 1, ... , 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, ... , 6 thus: 0, 3, 6, 2, 5, 1, 4.
(b) What about the numbers 0, 1, ... , 13?
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5. Prove that there exists a number divisible by 51000 with no zero digit.
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6. Find all integers x, y satisfying x2 + x = y4 + y3 + y2 + y.
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7. What is the maximum possible length of a sequence of natural numbers x1, x2, x3, ... such that xi ≤ 1998 for i ≥ 1, and xi = |xi-1 - xi-2| for i ≥3.
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8. 499 white rooks and a black king are placed on a 1000 x 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).
Prove or disprove each of (a) - (c).
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9. 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.
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10. 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.
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