From my search the summation cyc denote summations over permutations. But now there's a set involved, and I feel quite lost on how to put the meaning of it all together and how to proceed.
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The summation cyc denotes summation over the cyclic permutations, where you rotate $x,y,z$ around the circle. Have you tried the hint? Write out the three terms in the cyclic sum, put them over a common denominator, and see what happens. I am sure you will find some nice simplification. – Ross Millikan Oct 26 '17 at 19:17
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So, now what's a cyclic permutation? Is it the thing after the equal sign in part ii)? – Long Vuong Oct 26 '17 at 19:24
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That gives a definition, which is the usual one. If you have a triplet $(x,y,z)$ the cyclic sum includes $(x,y,z), (y,z,x), (z,x,y)$. It works for longer sequences, too. – Ross Millikan Oct 26 '17 at 19:58