Asian Pacific Mathematics Olympiad · 2001

Problems

Problem 1For a positive integer n, let S(n) be the sum of the digits in the decimal representation of n. A positive integer obtained by removing at least one digit from the right-hand end of n is called a stump of n. Let T(n) be the sum of all stumps of n. Prove that n=S(n)+9T(n).Solutions: 1
Problem 2Find the largest positive integer $N$ such that the number of integers in $\{1,2,\dots,N\}$ divisible by $3$ equals the number of integers divisible by $5$ or $7$ (or both).Solutions: 1
Problem 3Let two equal regular $n$ -gons $S$ and $T$ be located in the plane such that their intersection is a $2n$ -gon ( $n \ge 3$ ). The sides of $S$ are colored red and the sides of $T$ blue. Prove that the sum of the lengths of the blue sides of $S\cap T$ equals the sum of the lengths of its red sides.Solutions: 1
Problem 4A point in the plane with a Cartesian coordinate system is called a mixed point if one coordinate is rational and the other is irrational. Find all polynomials with real coefficients whose graphs contain no mixed point.Solutions: 1
Problem 5Find the greatest integer $n$ for which there are $n+4$ points $A,B,C,D,X_1,\dots,X_n$ in the plane with $AB\ne CD$ such that, for each $i=1,2,\dots,n$ , triangles $ABX_i$ and $CDX_i$ are congruent.Solutions: 1