| A1. an is the last digit of 1 + 2 + ... + n. Find a1 + a2 + ... + a1992. |
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| A2. Let f(x) = a1/(x + a1) + a2/(x + a2) + ... + an/(x + an), where ai are unequal positive reals. Find the sum of the lengths of the intervals in which f(x) ≥ 1. |
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| A3. ABC is an equilateral triangle with side 2. Show that any point P on the incircle satisfies PA2 + PB2 + PC2 = 5. Show also that the triangle with side lengths PA, PB, PC has area (√3)/4. |
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| B1. Let an, bn be two sequences of integers such that: (1) a0 = 0, b0 = 8; (2) an+2 = 2 an+1 - an + 2, bn+2 = 2 bn+1 - bn, (3) an2 + bn2 is a square for n > 0. Find at least two possible values for (a1992, b1992). |
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| B2. Construct a cyclic trapezium ABCD with AB parallel to CD, perpendicular distance h between AB and CD, and AB + CD = m. | |
| B3. Given a triangle ABC, take A' on the ray BA (on the opposite side of A to B) so that AA' = BC, and take A" on the ray CA (on the opposite side of A to C) so that AA" = BC. Similarly take B', B" on the rays CB, AB respectively with BB' = BB" = CA, and C', C" on the rays AB, CB. Show that the area of the hexagon A"A'B"B'C"C' is at least 13 times the area of the triangle ABC. |
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
11 July 2002