| 1. Let {x} denote the fractional part of x = x - [x]. The sequences x1, x2, x3, ... and y1, y2, y3, ... are such that lim {xn} = lim {yn} = 0. Is it true that lim {xn + yn} = 0? lim {xn - yn} = 0? |
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| 2. a1 + a2 + ... + an = 0, for some k we have aj ≤ 0 for j ≤ k and aj ≥ 0 for j > k. If ai are not all 0, show that a1 + 2a2 + 3a3 + ... + nan > 0. |
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| 3. Show that an integer = 7 mod 8 cannot be sum of three squares. |
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| 4. Let f(x) = 1 + 2/x. Put f1(x) = f(x), f2(x) = f(f1(x)), f3(x) = f(f2(x)), ... . Find the solutions to x = fn(x) for n > 0. |
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| 5. Let f(r) be the number of lattice points inside the circle radius r, center the origin. Show that limr→∞ f(r)/r2 exists and find it. If the limit is k, put g(r) = f(r) - kr2. Is it true that limr→∞ g(r)/rh = 0 for any h < 2? |
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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
23 September 2003
Last corrected/updated 23 Sep 03