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mean and the standard deviation

Source: 3-rd Taiwanese Mathematical Olympiad 1994

January 16, 2007
arithmetic sequencenumber theory unsolvednumber theory

Problem Statement

Prove that there are infinitely many positive integers nn with the following property: For any nn integers a1,a2,...,ana_{1},a_{2},...,a_{n} which form in arithmetic progression, both the mean and the standard deviation of the set {a1,a2,...,an}\{a_{1},a_{2},...,a_{n}\} are integers. Remark. The mean and standard deviation of the set {x1,x2,...,xn}\{x_{1},x_{2},...,x_{n}\} are defined by x‾=x1+x2+...+xnn\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n} and ∑(xi−x‾)2n\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}, respectively.
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