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Part of 2006 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Transposing numbers arranged in a circle

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

8/10/2011
The numbers 1,2,…,20061, 2,\ldots, 2006 are written around the circumference of a circle. A move consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up 1313 positions to the right of its initial position. lf the numbers 1,2,…,20061, 2,\ldots, 2006 are partitioned into 10031003 distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.
combinatorics unsolvedcombinatorics
Sequence x_{n+2} = gcd( x_{n+1} , x_{n} ) + 2006

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

8/10/2011
An infinite sequence x1,x2,…x_1,x_2,\ldots of positive integers satisfies xn+2=gcd⁡(xn+1,xn)+2006 x_{n+2}=\gcd(x_{n+1},x_n)+2006 for each positive integer nn. Does there exist such a sequence which contains exactly 10200610^{2006} distinct numbers?
number theorygreatest common divisornumber theory unsolved