Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.
Determine all sequences of real numbers which satisfy:
; for ;
Let an be a sequence of integers with values between and such that:
(i) Any two of the ’s are realtively prime,
(ii) Each is either a prime or a product of primes.
Determine the smallest possible values of to make sure that the sequence will contain a
Let be a cyclic quadrilateral such that the segments and are not parallel.
Consider the set of circles through and , and the set of circles through and .
Determine the set of points of tangency of circles in these two sets.
Let be a circle with radius and centre , and a fixed point in the interior of . Let and be perpendicular chords through . Consider the rectangles , and . Find the set of all points , and when moves around the whole circle.
Find the minimum positive integer such that there exists a function from the set of all integers to with the property that whenever .