

1. log_{8}2 = 0.2525 in base 8 (to 4 places of decimals). Find log_{8}4 in base 8 (to 4 places of decimals).


2. The Fibonacci sequence f_{1}, f_{2}, f_{3}, ... is defined by f_{1} = f_{2} = 1, f_{n+2} = f_{n+1} + f_{n}. Find all n such that f_{n} = n^{2}.


3. ABC is a triangle with ∠A = 90^{o}, ∠B = 60^{o}. The points A_{1}, B_{1}, C_{1} on BC, CA, AB respectively are such that A_{1}B_{1}C_{1} is equilateral and the perpendiculars (to BC at A_{1}, to CA at B_{1} and to AB at C_{1}) meet at a point P inside the triangle. Find the ratios PA_{1}:PB_{1}:PC_{1}.


4. p is a prime. Find all relatively prime positive integers m, n such that m/n + 1/p^{2} = (m + p)/(n + p).


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} (a^{k} + x^{k}), 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.

