MathDB

5

Part of 2013 IMC

Problems(2)

IMC 2013/1/5

Source: Imc 2013 Problem 5

8/8/2013
Does there exist a sequence (an)\displaystyle{\left( {{a_n}} \right)} of complex numbers such that for every positive integer p\displaystyle{p} we have that ∑n=1+∞anp\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} } converges if and only if p\displaystyle{p} is not a prime?
Proposed by Tomáš Bárta, Charles University, Prague.
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IMC 2013/2/5

Source: Imc 2013 Problem 10

8/9/2013
Consider a circular necklace with 2013\displaystyle{2013} beads. Each bead can be paintes either green or white. A painting of the necklace is called good if among any 21\displaystyle{21} successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.
Note. Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.
Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.
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