| 1. The sequence a0a1, a2, ... of non-negative integers is defined as follows. The first p-1 terms are 0, 1, 2, 3, ... , p-2. Then an is the least positive integer so that there is no arithmetic progression of length p in the first n+1 terms. If p is an odd prime, show that an is the number obtained by writing n in base p-1, then treating the result as a number in base p. For example, if p is 5, to get the 5th term one writes 5 as 11 in base 4, then treats this as a base 5 number to get 6. |
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| 2. A trigonometric map is any one of sin, cos, tan, arcsin, arccos and arctan. Show that given any positive rational number x, one can find a finite sequence of trigonometric maps which take 0 to x. [So we need to show that we can always find a sequence of trigonometric maps ti so that: x1 = t0(0), x2 = t1(x1), ... , xn = tn-1(xn-1), x = tn(xn).] |
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| 3. The circumcenter O of the triangle ABC does not lie on any side or median. Let the midpoints of BC, CA, AB be L, M, N respectively. Take P, Q, R on the rays OL, OM, ON respectively so that ∠OPA = ∠OAL, ∠OQB = ∠OBM and ∠ORC = ∠OCN. Show that AP, BQ and CR meet at a point. |
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| 4. a0, a1, a2, ... is an infinite sequence of integers such that an - am is divisible by n - m for all (unequal) n and m. For some polynomial p(x) we have p(n) > |an| for all n. Show that there is a polynomial q(x) such that q(n) = an for all n. |
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| 5. A graph with n points and k edges has no triangles. Show that it has a point P such that there are at most k(1 - 4k/n2) edges between points not joined to P (by an edge). |
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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
25 Aug 2002
Last corrected/updated 25 Aug 03