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public school (n.)

1570s, originally, in Britain, "a grammar school endowed for the benefit of the public," but most have evolved into boarding-schools for the well-to-do. From public (adj.) + school (n.1). The main modern meaning in U.S., "school (usually free) provided at public expense and run by local authorities," is attested from 1640s. 

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Definitions of public school from WordNet

public school (n.)
a tuition free school in the United States supported by taxes and controlled by a school board;
public school (n.)
private independent secondary school in Great Britain supported by endowment and tuition;
From wordnet.princeton.edu