* Winter Olympics start in Sochi in one year
* Putin wants Games to show Russia is a modern democracy
* Russia faces worries over costs, corruption, security
By Alexei Anishchuk
SOCHI, Russia, Feb 7 (Reuters) - Vladimir Putin fired a top Russian Olympic official on Thursday after publicly ridiculing him on a visit to half-finished sports complexes planned for a winter Olympics dogged by reports of corruption and construction delays.
The humiliation of Akhmed Bilalov, 42, stamped President Putin's authority over the 2014 Sochi Games and underlined the importance he attaches to a global event he hopes will show how far Russia has come since the Soviet Union collapsed in 1991.
In a vintage performance, reminiscent of an all-powerful tsar sweeping through town in imperial times, Putin became angry when he heard of the rising costs and construction delays at the ski-jump complex Bilalov has been involved in.
Unsmiling and sarcastic, Putin unceremoniously scolded Bilalov in front of television cameras at the Black Sea resort of Sochi on Wednesday, then sacked him as vice president of Russia's Olympic Committee.
"How is it possible that the vice-president of the Olympic Committee is delaying development?" Putin said after touring Sochi, which has become a huge construction site with the concrete carcasses of unfinished buildings hugging the skyline.
With Bilalov squirming in the background, he added: "Well done. You are really working well."
Bilalov's dismissal overshadowed a day of festivities as Russia unveiled huge diamond-shaped clocks in Moscow, Sochi and six other cities counting the days, hours, minutes and seconds to the opening of the Games a year from now.
Deputy Prime Minister Dmitry Kozak made clear Bilalov was also likely to lose his job on the board of the Resorts of the North Caucasus, a state firm created to develop luxury resorts.
"People who do not fulfil their obligations on such a scale cannot lead the Olympic movement in our country," Kozak told reporters in Sochi, a popular tourist destination for Russians in summer and winter.
In Sochi, the din of the heavy trucks and diggers did not stop as the day's countdown festivities began. Large areas are fenced off in the city centre and many of the roads are closed.
The ski resort of Krasnaya Polyana, which looks down on Sochi, is still a muddle of fences, cranes, trucks and unfinished buildings. There is little snow but plenty of mud.
Putin had warned officials on Wednesday not to let corruption push up the costs of the Games, already expected to reach $50 billion, or five times more than the initial price tag. This would make them the costliest Games so far.
The Olympics are a priority for Putin in his third term as president, a chance to show Russia is a modern democracy capable of organising global events, 13 years after he rose to power.
"The main thing is that no one steals anything, so there are no unexplained increases in costs," he said on Wednesday as he went rapidly from one venue to another.
Russia has been trying to shed its reputation for corruption, which has long put off foreign investors. But fears of foul play prompted then-President Dmitry Medvedev to order an investigation in 2010 into a senior Kremlin official accused of extorting bribes over Games construction work.
The price of building one less than 50-km (31-mile) stretch of road for the Games has been estimated at $7.5 billion, a figure so high that it fuelled more corruption allegations.
There is also a security threat from an Islamist insurgency raging nearby, concerns about environmental damage, allegations that some workers have been underpaid and residents are grumbling about the state of the city.
"It's good they are building new roads here, but the benefits of all of that will not be felt soon and people are getting upset," said Vladimir, a taxi driver in his mid-30s.
Approaching a new roundabout, he said: "This roundabout is a good example of how things work here. It's broad and smooth but it's surrounded by narrow roads, so there's no real point in it. There's traffic jams all around it anyway."
Thirteen official sites are being built, including a stadium that can house 40,000 people, plus facilities for ice hockey, skiing, snowboarding and skating. About 120,000 visitors are expected during the Games.
But beneath the slopes, Sochi has palm trees and a sub-tropical climate. The temperature on Thursday was around 13 degrees Celsius (55 Fahrenheit). Some Russians wonder why and how a Winter Olympics can be hosted by such a warm city.
The rising costs have also brought grumbles from wealthy Russian businessmen who have invested in the Games and want the government to provide more help with the funding.
At least half the money for the Games is coming from the state and private Russian businessmen or state-controlled companies are making up the rest, Russian Olympic officials say. (Writing by Alissa de Carbonnel and Timothy Heritage; editing by Ralph Boulton)