(CNN) -- Russian police detained dozens of people in an anti-terror sweep in Volgograd, the southern city where two suspected suicide bombings killed more than 30 people this week.
Thousands of police officers searched hundreds of people following the attacks on Sunday and Monday at a rail station and on a trolley bus just six weeks before Russia is due to host the Winter Olympics.
The death toll from the attacks increased to 34 when some of those hurt died from their injuries, state media reported.
Separately, Australia announced limits on the movement of its athletes for the Sochi Games.
The blasts in Volgograd intensified fears of terrorism following a threat earlier this year from a Chechen extremist group to use "maximum force" to disrupt the Olympics in February.
No group has claimed responsibility for the attacks described as suspected suicide bombings.
In his New Year's address, Russian President Vladimir Putin acknowledged problems in 2013 but said the Russian people had a lot to look forward to in 2014. He also had a message for terrorists.
"We bow our heads to the victims of violent terrorist attacks. We will continue the fight against terrorists certainly, fiercely and consistently until their complete annihilation," he said in the address on Russian TV.
In Volgograd, a strong security presence was on the streets leading to New Year's Eve celebrations.
Australia said Tuesday that none of its athletes competing in the Olympics will travel to Sochi using ground transportation in Russia -- they will only travel by air.
Australian athletes will only train and compete in Sochi and won't vacation elsewhere in Russia after the Games, said John Coates, the president of the Australian Olympic Committee.
"Families of athletes and all other participants of the Olympic Games, including media and spectators, should note the steps we are taking for the safety and welfare of our athletes," Coates said in a statement on the committee's website.
He nonetheless echoed comments from the International Olympic Committee expressing confidence that Russian authorities will do everything "to ensure the security of the athletes and all of the participants of the Olympic Games."
Key transit point
While security in Sochi and its surrounding area is considered to be very tight, the attacks in Volgograd, a major transportation hub for southern Russia, have raised concerns about the potential vulnerability of other areas of the country.
Volgograd is a main transit point for people traveling by train to Sochi, which is situated just over 400 miles (645 kilometers) to the southwest.
The number of people killed in an explosion at the Volgograd's main rail station Sunday now stands at 18, the state-run news agency RIA Novosti reported Tuesday, citing the Emergencies Ministry.
The toll from a blast on a trolleybus during the morning rush hour Monday has reached 16, the agency said, attributing the information to health officials.
Russian authorities have described both explosions as terrorist attacks and vowed to continue their "tough and consistent offensive" against extremists.
About 5,200 police officers are now on the ground in Volgograd for an anti-terrorism operation, Andrei Pilipchuk, a regional police official, said on Russian television.
Police are checking people's documents in all crowded areas of the city and have so far detained 87 people who put up resistance or didn't have documents allowing them to carry weapons, Pilipchuk said.
But it wasn't clear if any of those held had any suspected connection to the attacks or would face any charges.
A total of 1,500 buildings have been "examined" and more than 1,000 people have been searched in the operation, Pilipchuk said.
U.S. offers support
The United States has offered its "full support to the Russian government in security preparations for the Sochi Olympic Games," National Security Council spokeswoman Caitlin Hayden said in a statement Monday.
"We would welcome the opportunity for closer cooperation for the safety of the athletes, spectators, and other participants," Hayden said.
The U.S. State Department said American citizens planning to attend the Games should "be reminded that threats have been made against the Olympic Games and acts of terrorism, including bombings, continue to occur in Russia."
"This is an exciting, positive, happy international sporting event, but people going there do need to maintain vigilance and watch out for their own security and safety," Marie Harf, a State Department spokeswoman said at a regular briefing Monday.
Harf said the department was "very focused" on the security situation in Russia, but she declined to say whether any additional measures had been taken since the Volgograd bombings.
CNN's Diana Magnay and Arkady Irshenko contributed to this report.