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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 n-digit number with all its digits odd which is divisible by 5n .
  • 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 S1 of n+1 non-negative integers, a0, a1, ... , an we derive another sequence S2 with terms b0, b1, ... , bn, where bi is the number of terms preceding ai in S1 which are different from ai (so b0 = 0). Similarly, we derive S2 from S1 and so on. Show that if ai ≤ i for each i, then Sn = Sn+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 = MC2.
  • B2. Prove that for any positive reals x, y, z we have (2x+y+z)2/(2x2 + (y+z)2) + (2y+z+x)2/(2y2 + (z+x)2) + (2z+x+y)2/(2z2 + (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 (non-negative) difference between the two numbers at the adjacent vertices. If the starting numbers sum to 20032003, show that it is always possible to make a sequence of moves ending with zeros at every vertex.
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