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* Hong Kong airport reopens after four days of protests
* Airport administrator says flight movements will be affected
* Shutdown was blamed on anti-government protesters
* Cathay Pacific cancels more than 200 flights on Tuesday
* Protests post major challenge to Hong Kong and Chinese rule (Updates throughout with comments by Carrie Lam, details about flights and protests, Cathay Pacific's share price)
HONG KONG, Aug 13 (Reuters) - Hong Kong's airport reopened on Tuesday but its administrator warned that flight movements would still be affected, after China said protests that have swept the city over the past two months had begun to show the "sprouts of terrorism".
Some flights resumed but many others were cancelled. Hong Kong flag carrier Cathay Pacific said it had cancelled more than 200 flights into and out of the airport on Tuesday, according to its website.
The airport, one of the world's busiest, blamed demonstrators for halting flights on Monday. The exact trigger for the closure was not clear because protesters occupying the arrivals hall since Friday have been peaceful.
The airport was the latest focus of protests that began two months ago. The sometimes violent protests began as opposition to a now-suspended bill that would have allowed extradition to mainland China but have grown into wider calls for democracy.
The protests have plunged the Chinese-ruled territory into its most serious crisis in decades, presenting Chinese leader Xi Jinping with one of his biggest challenges since he came to power in 2012.
Analysts said the disruptions and protests in Hong Kong were unsettling Asian stock markets, and the Hang Seng index opened 1.1% lower on Tuesday.
Embattled Hong Kong Chief Executive Carrie Lam said on Tuesday the "lawbreaking activities in the name of freedom" were damaging the rule of law and that it could take a long time for the city to recover from the protests.
The protesters have been switching tactics in recent weeks and more than a dozen sit-ins were planned at hospitals in the city, according to social media posts on Tuesday.
Most of the protesters had left the airport shortly after midnight, with about 50 still there on Tuesday morning.
"Hong Kong International Airport will implement flight rescheduling today with flight movements expected to be affected," said a notice published on the Hong Kong International Airport's official mobile app on Tuesday.
Cathay Pacific said it would only operate a limited number of flights for connecting passengers. Airport flight boards showed the likes of Emirates Airline and Virgin Australia had flights scheduled to depart on Tuesday.
China said on Monday protests in the Asian financial hub had reached a critical juncture.
"Protesters have been frequently using extremely dangerous tools to attack the police in recent days, constituting serious crimes with sprouts of terrorism emerging," said Hong Kong and Macau Affairs office spokesman Yang Guang in Beijing.
Some Hong Kong legal experts say the official description of terrorism could lead to the use of anti-terror laws.
Protesters in turn say police have used excessive force, firing tear gas and bean bag pellets at close range, and are calling for an independent inquiry into the crisis.
Demonstrators say they are fighting the erosion of the "one country, two systems" arrangement enshrining some autonomy for Hong Kong when China took it back from Britain in 1997.
Hong Kong's airport is the 8th busiest by passenger traffic, handling 73 million passengers a year.
Shares in Cathay, which fell to a 10-year-low on Monday, continued their slide on Tuesday and were down more than 4.5% in morning trading.
The company is caught in crosswinds between Beijing and pro-democracy groups in Hong Kong after the Chinese civil aviation regulator demanded it suspend personnel who engaged in or supported the protests from staffing flights into its airspace.
The closure of the Hong Kong airport added to that pressure. A Reuters reporter saw more than 100 travellers queuing up at Cathay's ticketing counter early on Tuesday.
"The way to handle last night was chaotic," said Kate Flannery from Australia, who was travelling to Paris. "The airport authority didn't deal with the situation. I felt like I was walking around and nobody gave us information."
A Cathay customer officer at the airport, who declined to provide his name, said nearly all the airline's flights were full.
"It is possible that the airport authority will cancel more flights as they need to control the air traffic movements at the Hong Kong International Airport," he said. (Reporting by Brenda Goh and Felix Tam Editing by Michael Perry and Paul Tait)