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http://en.wikipedia.org/wiki/United_States_presidential_election,_1920#Republican_Party_nomination

On June 8, the Republican National Convention met in Chicago. The race was wide open, and soon the convention deadlocked between General Leonard Wood and Governor Frank O. Lowden of Illinois.

WI Lowden won the nomination and went on to run for president? This being 1920 I suspect any likeable Republican could win the race.
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