. Find the value of 
(1 mark)International Mathematical Olympiad Prelimary Selection Contest 2003-Hong Kong
1. Let . Find the value of (1 mark)
2. 15 students join a summer course. Every day, 3 students are on duty after school to clean the classroom. After the course, it was found that every pair of students have been on duty together exactly once. How many days does the course last for? (1 mark)
3. Find the number of pairs of consecutive integers in the set {1000,1001,1002,...,2000} such that no carrying is required when the two integeres are added. (1 mark)
4. A positive integer x is called a magic number if, when x is expressed in binary form, it possesses an even number of '1's. For example, the frist five magic numbers are 3,5,6,9 and 10. Find, in decimal notation, the sum of the first 2003 magic numbers. (1 mark)
5. A positive integer n is said to be increasing if, by reversing the digits of n, we get an integer larger than n. For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four digit positive integers are increasing? (1 mark)
6. The ratio of the sides of a triangle, which is inscribed in a circle of radius 
, is 3 : 5 : 7. Find the area of the triangle. (1 mark)
7. The number of apples produced by a group of farmers is less than 1000. It is known that they shared the apples in the following way. in turn, each farmer took from the collection of apples either exactly one-half or exactly one-third of the apples remaining in the collection. No apples were cut into pieces. After each farmer had taken his share, the rest was given to charity. Find the greatest number of farmers that could take part in the apple sharing. (1 mark)
8. Let a and b be positive integers such that 
and 
. Find ab. (1 mark)
9. Find an integer x such that 
. (1 mark)
10. Simplify (cos42o+cos102o+cos114o+cos174o)2 into a rational number. (1 mark)
11. On a certain planet there are 100 countries. They all agree to form unions, each with a maximum of 50 countries, and that each country will be joining a number of unions, so that every two different countries will belong to a same union. At least how many unions must be formed? (1 mark)
12. Find the last two digits of 7 x 19 x 31 x ... x 1999. (Here 7, 19, 31, ..., 1999 form an arithmetic sequence of common difference 12.) (1 mark)
13. ABCDEFGH is a cube in which ABCD is the top face, with vertices H, G, F and E directly below the vertices A, B, C and D respectively. A real number is assigned to each vertex. At each vertex, the average of the numbers in hte three adjacent vertices is then computed. The averages obtained at A, B, C, D, E, F, G, H are 1, 2, 3, 4, 5, 6, 7, 8 respectively. Find the number assigned to vertex F. (2 marks)
14. A regular 201-sided polygon is inscribed inside a circle of center C. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point C lying inside the triangle? (2 marks)
15. Given a rectangle ABCD, X and Y are respectively points on AB and BC. Suppose the areas of the triangles AXD, BXY and DYC are respectively 5, 4 and 3. Find the area of DXY. (2 marks)
16. Let ABC be an acute triangle, BC=5. E is a point on AC such that BE is perpendicular to AC, F is a point on AB such that AF=BF. moreover, BE=CF=4. Find the area of the triangle. (2 marks)
17. Given a triangle ABC, angle ABC=80o, angle ACB=70o and BC=2. A perpendicular line is drawn from A to BC, another perpendicular line drawn from B to AC. The two perpendicular lines meet at H. Find the length of AH. (2 marks)
18. Let A be a set containing only positive integers, and for any elements x and y in A, 
. Detemine at most how many elements A may contain. (2 marks)
19. A man chooses two positive integers m and n. He then defines a positive integers k to be good if a triangle with side length log m, log n and log k exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of mn. (3 marks)
20. The perimeter of triangle ABC, in which AB
AC, is 7 times the side length of BC. The inscribed circle of the triangle touches BC at E, and the diamtere DE of the circle is drawn, cutting the median from A to BC at F. Find DF/FE. (3 marks)