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Halep won 11 straight games at the start and beat Alisa Kleybanova of Russia 6-0, 6-2.
The 22-year-old Romanian has never been past the second round at Roland Garros, but she entered the clay-court major as one of the favorites following a run to the quarterfinals at the Australian Open.
Playing on Court Philippe Chatrier, the main stadium at Roland Garros, Halep raced through the first set and then won the first five games of the second. But she was broken while serving for the match the first time.
Halep finished with only six unforced errors, while Kleybanova had 29.
Date-Krumm, the oldest woman in the field at 43, had an up-and-down match against 24th-seeded Anastasia Pavlyuchenkova of Russia, eventually losing 6-3, 0-6, 6-2.
Date-Krumm is the third oldest player in French Open singles history; Martina Navratilova was 47 in 2004. She made her main-draw debut at Roland Garros in 1989 — before more than half of this year’s women’s field was even born.
No. 27 Svetlana Kuznetsova, the 2009 French Open champion, also advanced. The Russian beat Sofia Shapatava of Georgia 6-3, 6-1.
Later Tuesday, 2011 champion Li Na was playing her opening match. Seventh-seeded Andy Murray, the 2013 Wimbledon champion, was also scheduled to play.
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