| A1. Find the smallest positive integer N which is a multiple of 83 and is such that N2 has exactly 63 positive divisors. | |
| A2. Given two circles (neither inside the other) with different radii, a line L, and k > 0, show how to construct a line L' parallel to L so that L intersects the two circles in chords with total length k. | |
| A3. a, b, c, d are positive integers such that (a+b)2 + 2a + b = (c+d)2 + 2c + d. Show that a = c and b = d. Show that the same is true if a, b, c, d satisfy (a+b)2 + 3a + b = (c+d)2 + 3c + d. But show that there exist a, b, c, d such that (a+b)2 + 4a + b = (c+d)2 + 4c + d, but a ≠ c and b ≠ d. | |
| B1. Show that there are infinitely many primes in the arithmetic progression 3, 7, 11, 15, ... . | |
| B2. Given the triangle ABC, show how to find geometrically the point P such that ∠PAB = ∠PBC = ∠PCA. Express this angle in terms of ∠A, ∠B, ∠C using trigonometric functions. | |
| B3. For each positive integer n let S(n) be the set of complex numbers z such that |z| = 1 and (z + 1/z)n = 2n-1(zn + 1/zn). Find S(2), S(3), S(4). Find an upper bound for |S(n)| for n ≥ 5. |
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 March 2004
Last corrected/updated 25 Mar 04