| A1. Let Tn = 1 + 2 + ... + n = n(n+1)/2. Let Sn= 1/T1 + 1/T2 + ... + 1/Tn. Prove that 1/S1 + 1/S2 + ... + 1/S1996 > 1001. |
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| A2. Find an n in the range 100, 101, ... , 1997 such that n divides 2n + 2. |
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| A3. ABC is a triangle. The bisector of A meets the segment BC at X and the circumcircle at Y. Let rA = AX/AY. Define rB and rC similarly. Prove that rA/sin2A + rB/sin2B + rC/sin2C ≥ 3 with equality iff the triangle is equilateral. |
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| A4. P1 and P3 are fixed points. P2 lies on the line perpendicular to P1P3 through P3. The sequence P4, P5, P6, ... is defined inductively as follows: Pn+1 is the foot of the perpendicular from Pn to Pn-1Pn-2. Show that the sequence converges to a point P (whose position depends on P2). What is the locus of P as P2 varies? |
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| A5. n people are seated in a circle. A total of nk coins are distributed amongst the people, but not necessarily equally. A move is the transfer of a single coin between two adjacent people. Find an algorithm for making the minimum number of moves which result in everyone ending up with the same number of coins? |
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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 April 2002