XIV Asian Pacific Mathematics Olympiad March 2002

Time allowed: 4 hours No calculators are to be used Each question is worth 7 points

Problem 1.

Let a_{1}, a_{2}, a_{3}, …, a_{n} be a sequence of non-negative integers, where n is a positive integer. Let

A_{n} = \frac{a_{1} + a_{2} + ⋯ + a_{n}}{n} .

Prove that

a_{1}!a_{2}! … a_{n}! ≥ (⌊A_{n}⌋!)^{n} ,

where ⌊A_{n}⌋ is the greatest integer less than or equal to A_{n}, and a! = 1 × 2 × ⋯ × a for a ≥ 1 (and 0! = 1). When does equality hold?

Problem 2.

Find all positive integers a and b such that

\frac{a^{2} + b}{b^{2} − a} and \frac{b^{2} + a}{a^{2} − b}

are both integers.

Problem 3.

Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of ^{}C̸BP and ^{}B̸CQ such that triangle T RS is equilateral.

Problem 4.

Let x, y, z be positive numbers such that

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1.

Show that

\sqrt{x + yz} + \sqrt{y + zx} + \sqrt{z + xy} ≥ \sqrt{xyz} + \sqrt{x} + \sqrt{y} + \sqrt{z}.

Problem 5.

Let R denote the set of all real numbers. Find all functions f from R to R satisfying:

(i) there are only finitely many s in R such that f(s) = 0, and (ii) f(x^{4} + y) = x^{3}f(x) + f(f(y)) for all x, y in R.