Removing the Obvious

Number Theory Level 3

Does there exist a multi-set $A = \{ a_1, a_2, \ldots , a_{2016} \}$ of positive integers such that

$\sum a_i ^3 = (\sum a_i)^2$
$A \neq \{ 1, 2, \ldots , 2016 \}$
$a_1 \neq a_2?$

Note : Multi-set means that the elements could be repeated.

Yes No

1 solution

Jon Haussmann
Oct 6, 2016

Let $A$ be the multi-set where the number 1008 appears 1008 times, and the number 2016 appears 1008 times. Also, you can let $a_1 = 1008$ and $a_2 = 2016$ .

There are a lot of multi-sets that satisfy this condition. Can you find more in a similar "easy" manner?

Chung Kevin - 4 years, 8 months ago

Removing the Obvious

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