If no candidate wins a majority in the first round, which analysts are expecting, the top two finishers will face off in a runoff on January 26-27.
Before the second round of the previous presidential election in 2013, Babis told daily Hospodarske noviny (HN) he would vote for Zeman's rival Karel Schwarzenberg because Zeman continued with the capitalist mafia and corruption style of former president Vaclav Klaus.
Headlines like "Attack on Zeman" and "A scandalous incident" were splashed across the front pages of Saturday newspapers.
The first round of the presidential election will be held on Friday and Saturday. The woman, who Femen said was a Ukrainian citizen, was detained by security guards.
His political views are similar to those of other populist-minded eastern European Union leaders - including Hungary and Poland - that do not agree with Brussels over mandatory refugee quotas and other rules they see as efforts to limit national sovereignty.
He is also stridently anti-Muslim, having once called the 2015 migrant crisis "an organised invasion" of Europe and insisted Muslims were "impossible to integrate".
"It is a person who fights for our national interests, who is not afraid to present his clear opinion of Brussels and refugee quotas and is proud of his country", Babis said.
Drahos could not be more different. A mild-mannered liberal centrist whom critics have dubbed "wishy-washy", he has called for Prague to "play a more active role in the EU" and has backed the adoption of the euro.
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After submitting his vote in Prague Friday, Drahos said that the future president "should work in the interest of the pro-Western orientation of the Czech Republic".
"The president should work to unite society".
"Zeman's camp is very mobilized, but so are his opponents, because he's divided society so much", said Pavel Saradin, a political scientist at Palacky University in Olomouc.
Many voters remainied undecided until the last minute, with Prague archivist Marcela Riegerova saying she "ended up tossing a coin to decide between two candidates, and Drahos came out the victor".
The outcome may influence Czech Prime Minister Andrej Babis's chances of finally forming a cabinet as his first attempt to rule in a minority administration is likely to be rejected by parliament next week.
Milos Zeman once said he wanted "death for all abstainers and vegetarians", he has declared war-literally-to journalists and on environmental groups he said he would treat them "in the medieval way: he would burn them, urinate on them and I would throw salt at them".
But if one of the rivals wins, this will represent a huge change in the politics of this country that celebrates its 25th anniversary this year. A former head of the Czech Academy of Sciences, Drahos favours enhancing relations with the West.
The CTK news agency pegged turnout at 40% after day one of voting.