MathDB
Counting numbers on circle

Source: Moldavian TST_1

March 6, 2006
algorithminductioncombinatorics proposedcombinatorics

Problem Statement

Let mm circles intersect in points AA and BB. We write numbers using the following algorithm: we write 11 in points AA and BB, in every midpoint of the open arc ABAB we write 22, then between every two numbers written in the midpoint we write their sum and so on repeating nn times. Let r(n,m)r(n,m) be the number of appearances of the number nn writing all of them on our mm circles. a) Determine r(n,m)r(n,m); b) For n=2006n=2006, find the smallest mm for which r(n,m)r(n,m) is a perfect square. Example for half arc: 1−11-1; 1−2−11-2-1; 1−3−2−3−11-3-2-3-1; 1−4−3−5−2−5−3−4−11-4-3-5-2-5-3-4-1; 1−5−4−7−3−8−5−7−2−7−5−8−3−7−4−5−11-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1...
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