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In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens

x

and

x'

either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens

x = x_0, x_1, \ldots, x_n = x'

for some integer

n \geq 2

such that

x_{i-1}

and

x_i

know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

2

Problems(2)