THE 1993 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

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

Question 1

Let ABCD be a quadrilateral such that all sides have equal length and angle ABC is 60 deg. Let l be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF. Prove that CA^{2} = CM × CE.

Question 2

Find the total number of different integer values the function

f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x]

takes for real numbers x with 0 ≤ x ≤ 100.

Question 3

Let

f(x) = a_{n}x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{0} and
g(x) = c_{n+1}x^{n+1} + c_{n}x^{n} + ⋯ + c_{0}

be non-zero polynomials with real coefficients such that g(x) = (x + r)f(x) for some real number r. If a = max(|a_{n}|, …, |a_{0}|) and c = max(|c_{n+1}|, …, |c_{0}|), prove that \frac{a}{c} ≤ n + 1.

Question 4

Determine all positive integers n for which the equation

x^{n} + (2 + x)^{n} + (2 − x)^{n} = 0

has an integer as a solution.

Question 5

Let P_{1}, P_{2}, …, P_{1993} = P_{0} be distinct points in the xy-plane with the following properties:

(i) both coordinates of P_{i} are integers, for i = 1, 2, …, 1993;

(ii) there is no point other than P_{i} and P_{i+1} on the line segment joining P_{i} with P_{i+1} whose coordinates are both integers, for i = 0, 1, …, 1992.

Prove that for some i, 0 ≤ i ≤ 1992, there exists a point Q with coordinates (q_{x}, q_{y}) on the line segment joining P_{i} with P_{i+1} such that both 2q_{x} and 2q_{y} are odd integers.