Republican Donald Trump and Democrat Hillary Clinton scored bounce-back victories Tuesday on their home turf of New York, though the big delegate hauls they seized did little to slam the door shut on their chief challengers.

Vermont U.S. Sen. Bernie Sanders, who trails the former secretary of state by a nearly insurmountable deficit in the Democratic race, remains a formidable contender who has drawn huge crowds across the nation with his populist promises.

And Trump’s success in his home state did little to change the imposing electoral math he faces. Even with Tuesday’s sweeping victory, he retains only a narrow path to clinching the Republican nomination before the party’s July meeting. His opponents, Texas U.S. Sen. Ted Cruz and Ohio Gov. John Kasich, vow to force a contested convention.

Still, the victories were must-wins for both Trump and Clinton, who each suffered a string of stinging setbacks. Clinton had lost eight of the past nine contests to Sanders, while Cruz’s recent win in Wisconsin, and his dominance in the hunt for delegates in Georgia and other states, raised new questions about Trump’s durability.

Trump celebrated his victory at Trump Tower in Midtown Manhattan by predicting he would win enough delegates to avoid a convention fight.

“Senator Cruz is just about mathematically eliminated,” an exuberant Trump said in a brief victory speech, flanked by his family and advisers. “We’re really, really rocking.”

Clinton, too, cast herself as her party’s inevitable nominee. The race is in the homestretch, she said, “and victory is in sight.”

“Under the bright lights of New York, we have seen it’s not enough to diagnose the problems,” she told a cheering crowd in her adopted home state in a hotel blocks from the Trump Tower. “We have to solve the problems.”

Trump’s and Clinton’s successes in New York on Tuesday were never in doubt — only the size of their victories were in question. And both sought to run up their margins, with Clinton likely to net most of the 247 Democratic delegates up for grabs while Trump closed in on a potential sweep of the 95 his party offers.

It was also a symbolic victory for both in what has become an increasingly personal battle.

Born in Queens and rooted in Manhattan — he has his name on a skyscraper to prove it — Trump has campaigned as the quintessential New Yorker and assailed Cruz’s remarks criticizing his “New York values.” And Clinton, who represented the state in the U.S. Senate for eight years, has engaged in an escalating war of words with Sanders, a Brooklyn native.

Clinton and Trump each hoped a lopsided victory would send a message. Clinton wanted a chance to deliver a crippling blow to Sanders, who clashed with her in an edgy Brooklyn debate last week and drew more than 28,000 to a blockbuster event in Prospect Park days before the vote.

Trump risks losing the nomination to Kasich or Cruz if he fails to lock down 1,237 delegates in votes through June, and his campaign's failure to wrangle widespread support from delegates in Georgia over the weekend raised questions about its organizational skills.

Still, Trump’s and Clinton’s emboldened challengers have vowed to press on. Cruz and Sanders staged rallies Tuesday in Pennsylvania while Kasich visited Pennsylvania and Maryland. Those two states, along with Connecticut, Delaware and Rhode Island, hold votes on April 26. And polls show Trump and Clinton with solid leads in most of them.

Challengers sought niches

Trump’s campaign aimed to capture more than 50 percent of Tuesday’s vote by dominating New York City and locking up the conservative districts in the economically battered western and northern parts of the state.

Cruz and Kasich homed in on upstate New York voters as well, though both struggled to attract a following. Cruz, in particular, struggled to account for his “New York values” attack against Trump, which many of the billionaire’s supporters wore as a badge of honor.

At a rally in Philadelphia late Tuesday, the Texan unveiled a new slogan — “yes, we will” — and presented himself as the sole Republican who can implement his conservative campaign promises.

“Now is the time as Americans to once again reclaim that hope,” he said, making a not-so-subtle reference to President Barack Obama’s election mantra. “To take another giant leap for mankind.”

Kasich strategist John Weaver said the governor’s second-place finish in New York proves he’s best positioned to challenge Trump in Pennsylvania and the other April 26 states, and that a “vote for Cruz in these states is a vote for Trump.”

With little hope of an outright win, Sanders aimed for a symbolic victory by carrying one of New York's boroughs and winning over more minority voters. Like Clinton, he zeroed in on Brooklyn, the home to nearly 1 million registered Democrats, in hopes of rallying the party's liberal base to his side.

“I support Bernie with a passion,” said Tracy Wuischpard, a Brooklynite who attended a Sanders rally. “I would still vote for Hillary over a Republican, but we need bold changes. And I think he can win in November. I really do.”

But Sanders struggled to gain traction with women and minorities — the same blocs of voters who helped Clinton build a delegate lead with a string of victories in the South and West. In the closing days of the race, she held intimate rallies throughout the city, dancing with voters in Washington Heights and touring bakeries in Brooklyn.

“I could not be more excited to support Hillary,” said Brett Dakin, a Manhattanite who attended one of her closing rallies in Midtown. “I’ve literally been itching to vote for her. She’s the most qualified candidate. She represents my beliefs. And she will represent me well in the White House.”