The 32nd United States of America Mathematics Olympiad
2003年第32届美国数学奥林匹克 
A1. Show that for each n we can find an ndigit number with all its digits odd which is divisible by 5^{n} . A2. A convex polygon has all its sides and diagonals with rational length. It is dissected into smaller polygons by drawing all its diagonals. Show that the small polygons have all sides rational. A3. Given a sequence S_{1} of n+1 nonnegative integers, a_{0}, a_{1}, ... , a_{n} we derive another sequence S_{2} with terms b_{0}, b_{1}, ... , b_{n}, where b_{i} is the number of terms preceding a_{i} in S_{1} which are different from a_{i} (so b_{0} = 0). Similarly, we derive S_{2} from S_{1} and so on. Show that if a_{i} ≤ i for each i, then S_{n} = S_{n+1}. B1. ABC is a triangle. A circle through A and B meets the sides AC, BC at D, E respectively. The lines AB and DE meet at F. The lines BD and CF meet at M. Show that M is the midpoint of CF iff MB·MD = MC^{2}. B2. Prove that for any positive reals x, y, z we have (2x+y+z)^{2}/(2x^{2} + (y+z)^{2}) + (2y+z+x)^{2}/(2y^{2} + (z+x)^{2}) + (2z+x+y)^{2}/(2z^{2} + (x+y)^{2}) ≤ 8. B3. A positive integer is written at each vertex of a hexagon. A move is to replace a number by the (nonnegative) difference between the two numbers at the adjacent vertices. If the starting numbers sum to 2003^{2003}, show that it is always possible to make a sequence of moves ending with zeros at every vertex. 
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