| A1. The sides of a triangle form an arithmetic progression. The altitudes also form an arithmetic progression. Show that the triangle must be equilateral. |
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| A2. The positive integers a, b, c, d, p, q satisfy ad - bc = 1 and a/b > p/q > c/d. Show that q ≥ b + d and that if q = b + d, then p = a + c. |
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| A3. P is a fixed point in the plane. Show that amongst triangles ABC such that PA = 3, PB = 5, PC = 7, those with the largest perimeter have P as incenter. |
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| B1. Points A1, A2, ... , An are equally spaced on the side BC of the triangle ABC (so that BA1 = A1A2 = ... = An-1An = AnC). Similarly, points B1, B2, ... , Bn are equally spaced on the side CA, and points C1, C2, ... , Cn are equally spaced on the side AB. Show that (AA12 + AA22 + ... + AAn2 + BB12 + BB22 + ... + BBn2 + C12 + ... + CCn2) is a rational multiple of (AB2 + BC2 + CA2). |
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| B2. Let k3 = 2 and let x, y, z be any rational numbers such that x + y k + z k2 is non-zero. Show that there are rational numbers u, v, w such that (x + y k + z k2)(u + v k + w k2) = 1. |
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| B3. Let S be the collection of all sets of n distinct positive integers, with no three in arithmetic progression. Show that there is a member of S which has the largest sum of the inverses of its elements (you do not have to find it or to show that it is unique). |
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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
1 July 2002
Last corrected/updated 28 Nov 03