Mr Romney has been projected to win contests in Massachusetts, Virginia and Vermont, while Mr Santorum was forecast to win Tennessee and Oklahoma. Meanwhile, Newt Gingrich declared victory in Georgia. Also voting were Idaho, North Dakota and Alaska. The eventual nominee will challenge Barack Obama in November's election. 'Tortoise and rabbits' Front-runner Mr Romney, a former Massachusetts governor and private equity tycoon, has been riding a wave of momentum after poll wins in recent weeks. But former Pennsylvania Senator Santorum - who bills himself as the true conservative in the race and a candidate who can win over blue-collar voters - has been hoping to halt his rival's winning streak. In Atlanta, Mr Gingrich lashed out at "elites" in the Republican party and the news media who he said had worked aggressively to push him from the race. "There are lots of bunny rabbits that run through. I'm the tortoise, I just take one step at a time," he said. Texas Congressman Ron Paul, widely dismissed as a longshot candidate because of his libertarian-leaning views, hopes to notch up his first wins in Alaska and North Dakota's caucuses. At a rally in North Dakota, he gave no indication of whether he would stay in the race. "If we set a standard of individual liberty, the rest of the world will notice," Mr Paul said. "It's much easier to promote our cause through peaceful deeds than through war." Romney's renewed questions Rust-belt Ohio, a crucial swing state, is rated as Tuesday's most coveted prize as no Republican nominee has ever become president without winning the state in the general election. The BBC's North America editor, Mark Mardell, says there will be renewed questions asked about Mr Romney's appeal if he fails to win Ohio. Many believe it is inevitable that Mr Romney will eventually be chosen to take on Barack Obama, our correspondent says, but few feel hugely inspired by the former Massachusetts governor. This whole contest has been driven by the quest of the dominant conservative wing of the party to find somebody who is not Mr Romney, he adds, someone who is a true believer. A Washington Post/ABC News poll released on Tuesday showed 35% of Americans looked upon Mr Romney favourably, compared to 32% for Mr Paul, 23% for Mr Gingrich, and 32% for Mr Santorum. The Washington Post called the results "a sobering reminder for the party that the extended primary season has damaged the brand". Mr Romney's well-funded political machine has far outspent Mr Santorum in Ohio, chopping down his double-digit opinion poll lead in the Midwestern bellwether state. A devout Catholic who opposes abortion and gay marriage, Mr Santorum has sought to exploit lingering doubts among some conservatives about Mr Romney, a Mormon who governed in a liberal state. Obama's well wishes Mr Romney was on course for a convincing victory in Virginia, where Mr Santorum and Mr Gingrich failed to qualify for the ballot. His projected victory in Massachusetts came as no surprise as he was governor there for four years. He was also projected the winner in Vermont, which is next door to his political homeland. Analysts say Mr Romney has the advantage in Idaho, a state with a large Mormon population. Well wishes were extended to Mr Romney from an unexpected quarter on Tuesday - during a White House news conference. When asked what he made of Mr Romney branding him America's "most feckless president since Jimmy Carter", Barack Obama paused, smiled, and replied: "Good luck tonight," adding: "No, really". Out of the 1,144 delegates needed to secure the Republican presidential nomination at the party convention in August, more than 400 are at stake in Tuesday's contests. Ahead of the 10-state voting marathon, Mr Romney had a solid lead with 203 delegates, compared with 92 for Mr Santorum, 33 for Mr Gingrich and 25 for Mr Paul, according to the Associated Press.