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Problem 2 of Third round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

September 1, 2019
algebraprime numbers

Problem Statement

Let pp be a prime number and the decimal notation of 1p\frac{1}{p} is periodical with a length of the period 4k4k, 1p=0,a1a2…a4ka1a2…a4k…\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}… .Prove that a1+a3+...+a4k−1=a2+a4+...+a4ka_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}.
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