| A1. The triangle ABC has a, b, c = 29, 21, 20 respectively. The points D, E lie on the segment BC with BD = 8, DE = 12, EC = 9. Find ∠DAE. |
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| A2. A graph has n points. Each point has degree at most 3. If there is no edge between two points, then there is a third point joined to them both. What is the maximum possible value of n? What is the maximum if the graph contains a triangle? |
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| A3. Find all positive integer solutions to p(p+3) + q(q+3) = n(n+3), where p and q are primes. |
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| A4. Define the sequence a1, a2, a3, ... by a1 = a2 = a3 = 1, an+3 = (an+2an+1 + 2)/an. Show that all terms are integers. |
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| A5. Show that x/(1-x) + y/(1-y) + z/(1-z) ≥ 3(xyz)1/3/(1 - (xyz)1/3) for positive reals x, y, z all < 1. |
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| B1. For which n can we find a cyclic shift a1, a2, ... , an of 1, 2, 3, ... , n (ie i, i+1, i+2, ... , n, 1, 2, ... , i-1 for some i) and a permutation b1, b2, ... , bn of 1, 2, 3, ... , n such that 1 + a1 + b1 = 2 + a2 + b2 = ... = n + an + bn? |
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| B2. n = p·q·r·s, where p, q, r, s are distinct primes such that s = p + r, p(p + q + r + s) = r(s - q) and qs = 1 + qr + s. Find n. |
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| B3. Let Q be the rationals. Find all functions f : Q → Q such that f(x + f(y) ) = f(x) + y for all x, y. |
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| B4. Show that kn - [kn] = 1 - 1/kn, where k = 2 + √3. |
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| B5. The incircle of the triangle ABC touches BC at D and AC at E. The sides have integral lengths and |AD2 - BE2| ≤ 2. Show that AC = BC. |
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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
15 June 2003
Last updated/corrected 15 Dec 03