THE 1995 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours NO calculators are to be used. Each question is worth seven points.

Question 1

Determine all sequences of real numbers a_{1}, a_{2}, …, a_{1995} which satisfy:

2\sqrt{a_{n} − (n − 1)} ≥ a_{n+1} − (n − 1), for n = 1, 2, … 1994,

and

2\sqrt{a_{1995} − 1994} ≥ a_{1} + 1.

Question 2

Let a_{1}, a_{2}, …, a_{n} be a sequence of integers with values between 2 and 1995 such that:

(i) Any two of the a_{i}’s are realtively prime, (ii) Each a_{i} is either a prime or a product of primes.

Determine the smallest possible values of n to make sure that the sequence will contain a prime number.

Question 3

Let P QRS be a cyclic quadrilateral such that the segments P Q and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set A of points of tangency of circles in these two sets.

Question 4

Let C be a circle with radius R and centre O, and S a fixed point in the interior of C. Let AA′ and BB′ be perpendicular chords through S. Consider the rectangles SAMB, SBN′A′, SA′M′B′, and SB′NA. Find the set of all points M, N′, M′, and N when A moves around the whole circle.

Question 5

Find the minimum positive integer k such that there exists a function f from the set ℤ of all integers to {1, 2, … k} with the property that f(x) ≠ f(y) whenever |x − y| ∈ {5, 7, 12}.