By Cao Minh Quang.

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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. 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. 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?

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. 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. 60 ☺ The best problems from around the world Cao Minh Quang 2nd ASU 1968 problems 1. An octagon has equal angles. The lengths of the sides are all integers.

Show that you can form a sum s = b1a1 + ... + bnan with each bi +1 or -1, so that 0 ≤ s ≤ a1. 7. Prove that you can always draw a circle radius A/P inside a convex polygon with area A and perimeter P. 8. A graph has at least three vertices. Given any three vertices A, B, C of the graph we can find a path from A to B which does not go through C. Prove that we can find two disjoint paths from A to B. [A graph is a finite set of vertices such that each pair of distinct vertices has either zero or one edges joining the vertices.