pow:problem7f21
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| pow:problem7f21 [2021/12/10 15:51] – created mazur | pow:problem7f21 [2021/12/12 03:38] (current) – mazur | ||
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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' | ||
| + | some details are different. For details see the following link {{: | ||