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The 21st United States of America Mathematics Olympiad
1992年第21届美国数学奥林匹克
  1. Let an be the number written with 2n nines. For example, a0 = 9, a1 = 99, a2 = 9999. Let bn = ∏0n ai. Find the sum of the digits of bn.
  2. Let k = 1o. Show that ∑088 1/(cos nk cos(n+1)k ) = cos k/sin2k.
  3. A set of 11 distinct positive integers has the property that we can find a subset with sum n for any n between 1 and 1500 inclusive. What is the smallest possible value for the second largest element ?
  4. Three chords of a sphere are meet at a point X inside the sphere but are not coplanar. A sphere through an endpoint of each chord and X touches the sphere through the other endpoints and X. Show that the chords have equal length.
  5. A complex polynomial has degree 1992 and distinct zeros. Show that we can find complex numbers zn , such that if p1(z) = z - z1 and pn(z) = pn-1(z)2 - zn , then the polynomial divides p1992(z).
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