| 1. log82 = 0.2525 in base 8 (to 4 places of decimals). Find log84 in base 8 (to 4 places of decimals). |
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| 2. The Fibonacci sequence f1, f2, f3, ... is defined by f1 = f2 = 1, fn+2 = fn+1 + fn. Find all n such that fn = n2. |
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3. ABC is a triangle with ∠A = 90o, ∠B = 60o. The points A1, B1, C1 on BC, CA, AB respectively are such that A1B1C1 is equilateral and the perpendiculars (to BC at A1, to CA at B1 and to AB at C1) meet at a point P inside the triangle. Find the ratios PA1:PB1:PC1.
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| 4. p is a prime. Find all relatively prime positive integers m, n such that m/n + 1/p2 = (m + p)/(n + p). |
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| 5. f(x) is a polynomial of degree 2n. Show that all polynomials p(x), q(x) of degree at most n such that f(x)q(x) - p(x) has the form ∑2n<k≤3n (ak + xk), have the same p(x)/q(x). | |
| 6. f(x) is a real valued function defined for x ≥ 0 such that f(0) = 0, f(x+1) = f(x) + √x for all x, and f(x) < ½ f(x - ½) + ½ f(x + ½) for all x ≥ ½. Show that f(½) is uniquely determined. |
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
26 September 2003
Last corrected/updated 26 Sep 03