THE 1994 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

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

Question 1

Let f : ℝ → ℝ be a function such that

(i) For all x, y ∈ ℝ,

f(x) + f(y) + 1 ≥ f(x + y) ≥ f(x) + f(y),

(ii) For all x ∈ [0, 1), f(0) ≥ f(x),

(iii) −f(−1) = f(1) = 1.

Find all such functions f.

Question 2

Given a nondegenerate triangle ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH| < 3R.

Question 3

Let n be an integer of the form a^{2} + b^{2}, where a and b are relatively prime integers and such that if p is a prime, p ≤ \sqrt{n}, then p divides ab. Determine all such n.

Question 4

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

Question 5

You are given three lists A, B, and C. List A contains the numbers of the form 10^{k} in base 10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively:

A  B  C
10 1010  20
100 1100100  400
1000 1111101000 13000
⋮  ⋮  ⋮

Prove that for every integer n > 1, there is exactly one number in exactly one of the lists B or C that has exactly n digits.