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system in N*: a_1 + 2a_2 +... + na_n = 6n, 1/a_1+2/a_2+... +n/a_n=2+1/n

Source: Dutch IMO TST 2015 day 2 p2

August 30, 2019

system of equationsDiophantine Equationsdiophantinenumber theory

Problem Statement

Determine all positive integers

n

for which there exist positive integers

a_1,a_2, ..., a_n

with

a_1 + 2a_2 + 3a_3 +... + na_n = 6n

and

\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n