pow:problem7f21 [Department of Mathematics and Statistics, Binghamton University]

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<box 85% round orange| Problem 7 (due Monday, December 6)>

A universe consists of an infinite collection of galaxies $G_i$ indexed by the
integers. Each galaxy $G_i$ consists of a finite number $g_i$ of stars. Every star in galaxy $G_i$
is connected with every star in galaxy $G_{i-1}$, with every star in galaxy $G_{i+1}$, possibly
with some stars in galaxy $G_i$, and with no other stars. It is known that for every sufficiently
large $n$ the number $g_{-n}+g_{-n+1}+\ldots +g_{n-1}+g_{n}$ does not exceed $n^{3/2}$. Each star has mass equal to the arithmetic mean (i.e. the average) of the masses of all the stars connected with it. Prove that all stars have the same mass. 

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We received a partial solution from Pluto Wang, who showed that all stars in a given galaxy have the same mass,
and a complete solution from Ashton Keith. Ashton's solution follows the same ideas as our solution, though
some details are different. For details see the following link {{:pow:2021fproblem7.pdf|Solution}}.