| 1. Solve |||||x2-x-1| - 2| - 3| - 4| - 5| = x2 + x - 30. |
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| 2. Circle C center O touches externally circle C' center O'. A line touches C at A and C' at B. P is the midpoint of AB. Show that ∠OPO' = 90o. |
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| 3. Find non-negative integers a, b, c, d such that 5a + 6b + 7c + 11d = 1999. |
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| 4. An equilateral triangle side x has its vertices on the sides of a square side 1. What are the possible values of x? |
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| 5. xi are non-negative reals. x1 + x2 + ... + xn = s. Show that x1x2 + x2x3 + ... + xn-1xn ≤ s2/4. |
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| 6. S is any sequence of at least 3 positive integers. A move is to take any a, b in the sequence such that neither divides the other and replace them by gcd(a,b) and lcm(a,b). Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order. |
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
8 Oct 2003
Last corrected/updated 8 Oct 03