With less than three weeks to go, Russia's leader uses interview to hail February Games and offer tips for weight control

From persistent allegations of corruption to worries about gay rights and terrorism, Vladimir Putin has had few opportunities to present the Winter Olympics as he intended: a triumphant moment that has put Russia back on the map.

And so, with less than three weeks to go to the start of the Sochi Games, Putin gathered international television networks to a cosy mountain studio to put across the narrative of Sochi 2014 as a gay-friendly event in which every dollar has been accounted for and sport will be the winner.

The questioning, from journalists including the BBC's Andrew Marr and Bill Clinton's former spokesman George Stephanopoulos, now of ABC, was gentle enough. Putin's message, meanwhile, was strident, though not entirely convincing.

On corruption, Putin insisted there had been no systemic graft during the preparations for the Games, and said all allegations to the contrary were not backed up by solid facts.

"I do not see serious corruption instances for the moment, but there is a problem with overestimation of construction volumes," Putin said. He said some contractors had won tenders due to low bids which they subsequently inflated.

"This price increase, it is sometimes due to contractor's deliberate acts, and sometimes it is due to the fact that the professional valuation of necessary investments, especially in mountain conditions, for a mountain cluster, are not efficient enough."

Previously the official figure for spending on Sochi had been put at about $51bn (£31bn), four times the planned cost and more than either the London or Beijing summer Olympics. One critical report puts the corruption estimate at $30bn – a figure greater than the GDP of more than half the world's countries.

IOC member Gian Franco Kasper has said that as much as one-third of the $51bn price tag – the largest in the history of the Olympics – has been siphoned off. One road alone cost around £6bn – more than the entire cost of the previous Winter Olympics in Vancouver.

But Putin told the journalists that the real cost of the Sochi Olympics was just 214bn roubles (about £4bn), and that all the other spending involved necessary improvements to transport infrastructure, and creating Russia's first modern holiday resort. He said Kasper's words had been taken out of context, and added that no one had proof of corruption.

"If anyone has such information, give it to us, please. I repeat once again, we will be grateful. But so far there was nothing but talks," he said.

The Olympics are a pet project for Putin, who has a residence in Sochi and has personally overseen preparations since Russia won the right to host the Games. He admitted that he thought the Olympics were important as part of his project to boost national pride: "There is also a certain psychological aspect here and we can talk about it directly, without any embarrassment or pretence.

"After the collapse of the Soviet Union, after the dark and, let us be honest, bloody events in the Caucasus, the public attitude in Russia became very negative and pessimistic. We have to pull ourselves together and realise that we can deliver large-scale projects on time and with high standards."

On gay rights, he was equally firm. Russia's new law banning homosexual propaganda is not discriminatory and is "only" about restricting the "propaganda of paedophilia and homosexuality" among children. He said the Russian law was designed "to protect children and future demographic development" and that if gay people were successful, he wished them all the best. "When they achieve great results, such as, for instance Elton Johnachieves, who is an extraordinary person, a distinguished musician, millions of our people sincerely love him with no regard to his sexual orientation."

Putin took only a few questions one-on-one from the foreign journalists, preferring a round-table format for the main interview that provided little opportunity for follow-up questions or real probing. The BBC was joined by other international channels and two reporters from Russian state television, who asked softball questions and even complained when the foreign journalists raised the gay rights issue.

The Chinese television reporter present told Putin he was very popular in China and suggested he should consider playing "tough guys" in action movies when he retired. Putin said that was unlikely but that he would probably continue to play ice hockey.

But, when asked the secret to his own good health, the Russian president had a few tips: "How to control one's weight? Don't overeat. How to stay in shape? Do sports. There are no magic pills here. I do a bit of sports every day.

"Yesterday I skied here until 1.30am, this morning I exercised in the gym, and I swim 1000 metres almost every day. That's nothing special, but I do it regularly. You know the saying: the chicken pecks grain after grain."

When Marr got Putin one-on-one briefly, he again raised the gay issue. Putin insisted he had no personal problem with gay people: "If you want to know what I personally think about this, a person's sexual orientation actually does not make any difference to me. I know a few of them, and I am on friendly terms with some of them."

Oddly, Marr devoted half of his two minutes with the Russian president to the Scottish independence referendum. Putin said it was "an internal issue" but that if Scotland did vote yes, he might consider inviting the newly independent country into the Customs Union, Russia's new trade bloc.

"I can't rule it out," said Putin with a smirk.

Winter Olympics 2014: the problems

Price tag At £31bn it has cost more than any previous winter or summer Games. Many of the biggest tenders have gone to businessmen close to the Kremlin.

Security The Games are taking place close to the volatile North Caucasus region. A double suicide attack in Volgograd just before new year highlighted the dangers, and more than 40,000 security personnel will be on duty. Vladimir Putin said he hoped security would be "effective" but "not intrusive or too conspicuous".

Gay rights Could gay athletes or spectators could fall foul of Russia's new laws on homosexual propaganda? At the weekend, a gay activist was detained after waving a rainbow flag at the Olympic torch rally in city of Voronezh.

The snow Sochi, one of Russia's few subtropical cities, has none, and today's temperature is expected to hit 13 degrees. But the mountain resort of Krasnaya Polyana is just a few kilometres away.