THE 1989 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

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

Question 1

Let x_{1}, x_{2}, …, x_{n} be positive real numbers, and let

S = x_{1} + x_{2} + ⋯ + x_{n}.

Prove that

(1 + x_{1})(1 + x_{2}) ⋯ (1 + x_{n}) ≤ 1 + S + \frac{S^{2}}{2!} + \frac{S^{3}}{3!} + ⋯ + \frac{S^{n}}{n!} .

Question 2

Prove that the equation

6(6a^{2} + 3b^{2} + c^{2}) = 5n^{2}

has no solutions in integers except a = b = c = n = 0.

Question 3

Let A_{1}, A_{2}, A_{3} be three points in the plane, and for convenience, let A_{4} = A_{1}, A_{5} = A_{2}. For n = 1, 2, and 3, suppose that B_{n} is the midpoint of A_{n}A_{n+1}, and suppose that C_{n} is the midpoint of A_{n}B_{n}. Suppose that A_{n}C_{n+1} and B_{n}A_{n+2} meet at D_{n}, and that A_{n}B_{n+1} and C_{n}A_{n+2} meet at E_{n}. Calculate the ratio of the area of triangle D_{1}D_{2}D_{3} to the area of triangle E_{1}E_{2}E_{3}.

Question 4

Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least

4m · \frac{(m − \frac{n^{2}}{4} )}{3n}

triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.

Question 5

Determine all functions f from the reals to the reals for which

(1) f(x) is strictly increasing,

(2) f(x) + g(x) = 2x for all real x, where g(x) is the composition inverse function to f(x). (Note: f and g are said to be composition inverses if f(g(x)) = x and g(f(x)) = x for all real x.)