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infinite n such that C_n^{S_n} is even, where S_n =[1+1/2+..+1/n]

Source: China Northern MO 2015 grade 11 p7 CNMO

May 5, 2024
number theory

Problem Statement

Use [x][x] to represent the greatest integer no more than a real number xx. Let Sn=[1+12+13+...+1n],  (n=1,2,..,)S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,) Prove that there are infinitely many nn such that CnSnC_n^{S_n} is an even number.
PS. Attached is the original wording which forgets left [ . I hope it is ok where I put it.
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