MathDB

6

Part of 2002 IMO Shortlist

Problems(2)

IMO ShortList 2002, algebra problem 6

Source: IMO ShortList 2002, algebra problem 6

9/28/2004
Let AA be a non-empty set of positive integers. Suppose that there are positive integers b1,…bnb_1,\ldots b_n and c1,…,cnc_1,\ldots,c_n such that - for each ii the set biA+ci={bia+ci ⁣:a∈A}b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\} is a subset of AA, and - the sets biA+cib_iA+c_i and bjA+cjb_jA+c_j are disjoint whenever i≠ji\ne j Prove that 1b1+ … +1bn≤1.{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.
algebraInequalityreciprocal sumIMO ShortlistAdditive Number Theory
IMO ShortList 2002, combinatorics problem 6

Source: IMO ShortList 2002, combinatorics problem 6; 54th Polish 2003

9/28/2004
Let nn be an even positive integer. Show that there is a permutation (x1,x2,…,xn)\left(x_{1},x_{2},\ldots,x_{n}\right) of (1, 2, …,n)\left(1,\,2,\,\ldots,n\right) such that for every i∈{1, 2, ..., n}i\in\left\{1,\ 2,\ ...,\ n\right\}, the number xi+1x_{i+1} is one of the numbers 2xi2x_{i}, 2xi−12x_{i}-1, 2xi−n2x_{i}-n, 2xi−n−12x_{i}-n-1. Hereby, we use the cyclic subscript convention, so that xn+1x_{n+1} means x1x_{1}.
graph theorycombinatoricsEulerian pathIMO Shortlist