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Find all numbers that can be represented in the given form

Source: Balkan MO ShortList 2010 A4

April 5, 2020

algebra unsolvedBalkan Mathematics Olympiad

Problem Statement

Let

n>2

be a positive integer. Consider all numbers

S

of the form \begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*} with

k>1

and

a_i

begin positive integers such that

a_1+a_2+ \ldots + a_k=n

. Determine all the numbers that can be represented in the given form.