World number one Ash Barty won her first WTA Tour match in almost a year at the Yarra Valley Classic, while Naomi Osaka got her Gippsland Trophy campaign going with a victory.
Barty sat out the vast majority of the 2020 season due to the coronavirus pandemic but marked her comeback with a 6-3 6-3 success over Ana Bogdan at Melbourne Park.
The Australian will face Marie Bouzkova next after the Czech defeated Aliona Bolsova on Tuesday.
Garbine Muguruza raced past Alison Van Uytvanck 6-2 6-0 and she was joined in the third round by Petra Kvitova, who battled to beat Venus Williams 7-6 (8-6) 7-5.
Reigning Australian Open champion Sofia Kenin, who was 7-5 up when Camila Giorgi retired from their match, is also through.
In her first match since winning the US Open in September, Osaka cruised to a 6-2 6-2 success against Alize Cornet and will meet Katie Boulter after the Briton came from a set down to beat Coco Gauff 3-6 7-5 6-2.
Iga Swiatek was also making her first appearance since winning a major title – the French Open last October – and after a slow start won 10 of the final 11 games to beat Kaja Juvan 2-6 6-2 6-1.
Aryna Sabalenka ended 2020 with back-to-back titles in Ostrava and Linz, before kicking off the new year with another trophy in Abu Dhabi. However, her run of 15 straight victories ended with a 1-6 6-2 1-6 loss to Kaia Kanepi.
Bianca Andreescu was due to gear up for the Australian Open by contesting the Grampians Trophy this week but withdrew from the competition to focus on training instead.
World number one Swiatek sees off Xiyu in Madrid opener
World number one Iga Swiatek overcame Wang Xiyu with the minimum of fuss on Thursday to progress to the third round of the Madrid Open.
Swiatek has yet to win the title in Madrid, which is the only clay-court European event at which the Pole has not triumphed, but she got off to a flying start with a 6-1 6-4 victory over Xiyu.
"I love this place. I got to know the city a little better last year," Swiatek said. "So this time I feel more comfortable around."
Next up for the 22-year-old is a third-round tie against 27th seed Sorana Cirstea on Saturday.
Data Debrief
Swiatek has registered 150 match wins on the WTA Tour since the start of 2022. Elena Rybakina and Coco Gauff are the next best with 108 wins each.
She also became the first player born this century to claim 50 combined match wins in Madrid, Rome and Roland-Garros, since the inaugural women's singles event at the Madrid Open in 2009.
Dominant Gauff double bagels Rus at Madrid Open
Coco Gauff made a flying start to her Madrid Open campaign on Thursday, recording the first double bagel main-draw win of her career as she cruised past Arantxa Rus to reach the round of 32.
Gauff needed just 51 minutes to see off 33-year-old Rus as she booked a meeting with Ukraine's Dayana Yamstremska for Saturday.
The US Open champion lost just 18 points throughout the match – eight in the opener and 10 in the second set – as she put any doubts over her recent form to bed in emphatic fashion.
Gauff's power proved too much for Rus in the opener, the third seed winning 88 per cent of points behind her first serve and converting three of four break points to go a set up in just 23 minutes.
Rus brought up four break points midway through the second set, but when she failed to convert them, Gauff punished her in ruthless fashion by reeling off another three games then finishing things with a terrific forehand winner.
Speaking after her win, the American said: "I played really well today. It was just one of those days for me… 6-0 6-0, I've never done that before, I'll probably never do it again."
Data Debrief: Gauff matches Azarenka, Halep feat
Gauff is just the third player to register a 6-0 6-0 win in the women's singles draw at the Madrid Open, following in the footsteps of two former world number ones in Victoria Azarenka and Simona Halep.
The American, who only turned 20 last month, is also the youngest player to double bagel an opponent in any main-draw match on the WTA Tour since Iga Swiatek did so against Karolina Pliskova in the 2021 Italian Open final in Rome.
Nadal cruises past teen Blanch to tee up De Minaur rematch in Madrid
Darwin Blanch presented little challenge for Rafael Nadal, who dropped just one game in a convincing 6-1 6-0 triumph at the Madrid Open.
Nadal is feeling his way back to fitness, but 16-year-old qualifier Blanch offered little resistance to the 22-time major champion on Thursday.
Making his farewell appearance at the Madrid Open, where he has won five titles, Nadal needed just 64 minutes to progress to round two.
"I think today I played against an opponent with a great future in front [of him], but today, still making mistakes," said Nadal.
"I just tried to be there, be solid all the time without taking a lot of risks. It worked well. I'm happy to be through and I wish him all the very best for the future."
Nadal's reward is a rematch with Alex de Minaur, who he lost to last week in Barcelona.
Data Debrief
This game saw the largest age gap (21 years, 117 days) between two opponents in the history of ATP Masters 1000 events.
Nadal has now registered 57 wins at the Madrid Open, which is 20 clear of any other player.
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