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^\circ.
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 \times CE.

Question 2
Find the total number of different integer values the function
f(x)=[x] + [2x]+\left[\frac{5x}{3}\right]+[3x] + [4x]
takes for real numbers x with 0 = x = 100.
Question 3
Let
f(x)= a_nx^n + a_{n-1}x^{n-1} +...+ a_0 and
•••
g(x)= c_{n+1}x_{n+1} + c_nx_n +...+ c_0
•••
be non-zero polynomials with real coeffcients 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 \displaystyle \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 \leq i \leq 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.

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