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2 runners

Source: Indian IMOTC 2004 Day 3 Problem 3

September 23, 2005
number theoryrelatively primecombinatorics unsolvedcombinatorics

Problem Statement

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds v1v_1 and v2v_2 respectively, where v1v_1 and v2v_2 are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time tt, at least one of the runners is at a distance not more than [v1+v22]v1+v2\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2} units from the starting point. (b) there is a time tt such that both the runners are at least [v1+v22]v1+v2\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2} units away from the starting point. (All disstances are measured along the track). [x][x] is the greatest integer function.
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