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For Mr. Santorum, Tuesday’s contests provided him with another opportunity to regain some of the momentum he appeared to have at the beginning of February, when he won caucuses in Colorado and Minnesota and a nonbinding primary in Missouri.

But his wins in the heartland masked the problems he has had running a national campaign.

He and Mr. Gingrich failed to qualify for the Virginia ballot, and Mr. Santorum’s appearance in Steubenville was an interesting choice, given that it is located in one of the congressional districts where Mr. Santorum’s campaign infrastructure failed him. He did not file the number of signatures Ohio party rules require to receive delegates to the national convention, based on the primary’s results.

In other words, even if he won the district, he would not have benefited in the delegate count that determines the nominee.

John Brabender, Mr. Santorum’s longtime political adviser, said they saw good signs from the vote.

“We won Tennessee going away; I think we won every county. We won Oklahoma going away. We just won North Dakota, and I think the Romney people came in third,” he told reporters. “All I know is that after spending close to $10 million here in Ohio, right now it is neck-and-neck, it is going to be close to a tie, and somebody will win, but not by much. I don’t know how they declare that anything but a disappointment.”

Each of the candidates has vowed to stay in the race at least through this month, and as if to underscore that commitment, Mr. Gingrich and Mr. Paul had already released public campaign schedules showing them traveling to Kansas ahead of that state’s Saturday caucuses.

With the exception of Mr. Paul, the other candidates also made video addresses to the American Israel Public Affairs Committee’s meeting in Washington, where they embraced Israel and leveled stiff criticism at the Obama administration’s approach toward Iran’s nuclear program.

Stephen Dinan reported from Washington.