Asian Pacific Mathematics Olympiad · 2000

Problems

Problem 1Compute the sum $S = \sum_{i=0}^{101} \frac{x_i^3}{1 - 3x_i + 3x_i^2}$ for $x_i = \frac{i}{101}$ .Solutions: 1
Problem 2Given a triangular arrangement of $9$ circles along the perimeter of a triangle, with $4$ circles on each of the three sides (three vertex circles shared by adjacent sides and two interior circles on each side), each of the numbers $1, 2, \ldots, 9$ is to be written into one of these circles, so that each circle contains exactly one of these numbers and: (i) the sums of the four numbers on each side of the triangle are equal; (ii) the sums of the squares of the four numbers on each side of the triangle are equal. Find all ways in which this can be done.Solutions: 1
Problem 3Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$ . Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$ , respectively, and $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced. Prove that $QO$ is perpendicular to $BC$ .Solutions: 2
Problem 4Let $n, k$ be given positive integers with $n > k$ . Prove that $\frac{1}{n+1} \cdot \frac{n^n}{k^k (n-k)^{n-k}} < \frac{n!}{k!(n-k)!} < \frac{n^n}{k^k (n-k)^{n-k}}$ .Solutions: 1
Problem 5Given a permutation $(a_0, a_1, \ldots, a_n)$ of the sequence $0, 1, \ldots, n$ . A transposition of $a_i$ with $a_j$ is called legal if $a_i = 0$ for $i > 0$ , and $a_{i-1} + 1 = a_j$ . The permutation $(a_0, a_1, \ldots, a_n)$ is called regular if after a number of legal transpositions it becomes $(1, 2, \ldots, n, 0)$ . For which numbers $n$ is the permutation $(1, n, n-1, \ldots, 3, 2, 0)$ regular?Solutions: 1