| 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. |
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| 2. Let k = 1o. Show that ∑088 1/(cos nk cos(n+1)k ) = cos k/sin2k. |
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| 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? |
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| 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. |
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| 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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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
5 May 2002