| A1. The triangle ABC has AB = 5, the medians from A and B are perpendicular and the area is 18. Find the lengths of the other two sides. |
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| A2. Find integers m and n such that n2 is a multiple of m, m3 is a multiple of n2, n4 is a multiple of m3, m5 is a multiple of n4, but n6 is not a multiple of m5. |
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| A3. Show that there is no positive integer of 1989 digits, at least three of them 5, such that the sum of the digits is the same as the product of the digits. |
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| B1. Find a positive integer n with decimal expansion amam-1...a0 such that a1a0amam-1...a20 = 2n. |
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B2. C1 and C2 are two circles of radius 1 which touch at the center of a circle C of radius 2. C3 is a circle inside C which touches C, C1 and C2. C4 is a circle inside C which touches C, C1 and C3. Show that the centers of C, C1, C3 and C4 form a rectangle.
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B3. How many paths are there from A to B which do not pass through any vertex twice and which move only downwards or sideways, never up?
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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
21 February 2004
Last corrected/updated 21 Feb 04