Asian Pacific Mathematics Olympiad · 1991

Problems

Problem 1Let G be the centroid of triangle ABC and M the midpoint of BC. Points X on AB and Y on AC are such that X, Y, G are collinear and XY is parallel to BC. Suppose XC and GB meet at Q, while YB and GC meet at P. Prove that triangle MPQ is similar to triangle ABC.Solutions: 1
Problem 2Suppose 997 points are given in the plane. Every segment joining two points has its midpoint coloured red. Prove that there are at least 1991 red points. Find a special case with exactly 1991 red points.Solutions: 1
Problem 3Let a_1,a_2,...,a_n and b_1,b_2,...,b_n be positive real numbers with a_1+...+a_n=b_1+...+b_n. Prove that $\sum_{i=1}^n \frac{a_i^2}{a_i+b_i} \ge \frac{a_1+\cdots+a_n}{2}$ .Solutions: 1
Problem 4During a break, n children sit in a circle. A teacher chooses one child and gives a candy, skips the next child and gives one to the following child, then skips 2 children, then 3, and so on. For which n will every child eventually receive at least one candy?Solutions: 1
Problem 5Two tangent circles and a point P on their common tangent perpendicular to the line joining their centres are given. Construct with ruler and compass all circles tangent to the two given circles and passing through P.Solutions: 1