Asian Pacific Mathematics Olympiad · 2004

Problems

Problem 1Determine all finite nonempty sets $S$ of positive integers satisfying $\frac{i+j}{\gcd(i,j)}\in S$ for all $i,j\in S$ .Solutions: 1
Problem 2Let $O$ be the circumcentre and $H$ the orthocentre of an acute triangle $ABC$ . Prove that the area of one of $\triangle AOH$ , $\triangle BOH$ , and $\triangle COH$ equals the sum of the areas of the other two.Solutions: 1
Problem 3Let S be a set of 2004 points in the plane, no three collinear, and let L be the set of all lines determined by pairs of points of S. A line separates two points when they lie on opposite sides of it and neither lies on the line. Prove that S can be colored with at most two colors so that two points have the same color exactly when an odd number of lines in L separates them.Solutions: 1
Problem 4For a real number x, let floor(x) be the greatest integer not exceeding x. Prove that floor((n-1)! divided by n(n+1)) is even for every positive integer n.Solutions: 1
Problem 5Prove that (a squared plus 2)(b squared plus 2)(c squared plus 2) is at least 9(ab+bc+ca) for all positive real numbers a,b,c.Solutions: 1