RIO DE JANEIRO — When the municipal velodrome was inaugurated here for the 2007 Pan American Games, it was the first training space for competitive indoor cyclists in the city. The wooden floors of the $7 million structure were the best quality in the country, as velodromes in the Brazilian states of São Paulo and Paraná were made of concrete.

Athletes took to the space, and Brazilians began winning medals. A cycling school in the velodrome that trained about 100 children from low-income communities had identified a girl with “absurd potential,” said Claudio Santos, the president of the Rio de Janeiro state cycling federation.

Nearly six years later, the velodrome was demolished, and now a new one is being built as Rio de Janeiro spends billions construcing facilities to host the Summer Olympics in nearly a year with the opening ceremony on Aug. 5, 2016. This city of 6.4 million has undergone a decade of face-changing transformations to prepare for mega-sporting events such as the Olympics, Pan American Games and soccer’s World Cup last summer. However, the process has been anything but efficient, with venues being constructed only to be razed or renovated in preparation for next year’s Games.

“On the eve of the Olympics, we have had no space for training,” Santos said.

Brazilian cyclists find themselves training in Switzerland for the Olympics that will be held in their own country. Rio de Janeiro Mayor Eduardo Paes had once dismissed the notion of demolishing the velodrome as wasteful, but he changed course following his re-election in 2012, saying that the cost to rehab it to meet Olympic standards would be about the same as building a new one.

Although Brazil’s strong economy and political stability was a selling point in hosting the first Olympics in South America, a sharp economic downturn and widespread corruption scandal have shifted attitudes toward hosting big events.

Many cariocas, as locals are called, have embraced key infrastructure projects such as expanded public transportation, but others question the planning behind the construction boom, not to mention the ballooning tab the events pass on to taxpayers.

‘NEXT BIG THING’

Brazilian authorities said this year that the total cost of the 2016 Games infrastructure and legacy projects will be at least 37.7 billion reais, about $11 billion. However, a quarter of the construction projects presented by budget authorities in January still did not have a price tag.

Rio de Janeiro had been bidding to host worldwide sporting events since 1997, when it presented a candidacy to host the 2004 Olympics. It would make two more bids for the Summer Olympics before it won that right for the 2016 Games, and in the meantime was also selected to host the 2007 Pan American Games and the 2014 World Cup.

The Summer Olympics had long been held in developed countries of the Northern Hemisphere, and the turn of the millennium marked a change when Beijing hosted the event in 2008.

That was a time — in the words of Pedro da Luz Moreira, the president of the Brazilian Architects Institute — that Brazil was “the next big thing.” It boasted a growing middle class and proudly identified as a BRIC, an acronym coined by a Goldman Sachs economist to refer to emerging economies and political powers Brazil, Russia, India and China.

In its successful 2009 bid-book to host the Summer Olympics, authors describe the country as “well-positioned as a result of its long-term growth, supported by proven economic policies.”

Rio’s architects were optimistic about the sporting events, Moreira said, as they saw a chance to take advantage of an injection of resources to carry out projects with a firm deadline in a city whose development had long been haphazard and geographically segmented.

But he said that optimism faded as architects saw commercial interests winning out over city planning and a reversion to the old Brazilian habit of improvisation.

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