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number theory: existence of some numbers

Source: Romanian District Olympiad 2015, Grade IX, Problem 3

September 25, 2018
number theorycontests

Problem Statement

Let m,n m, n natural numbers with m≥2,n≥3. m\ge 2,n\ge 3. Prove that there exist m m distinct multiples of n−1, n-1, namely, a1,a2,a3,...,am, a_1,a_2,a_3,...,a_m, such that: 1n=∑i=1m(−1)i−1ai. \frac{1}{n} =\sum_{i=1}^m \frac{(-1)^{i-1}}{a_i} .
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