She said equality for women was "at the heart of the Scottish government".
More details about the competition, which is being run by the Young Scot organisation, will be released at a later date.
Ms Sturgeon has also called on women in leadership roles to follow her example and become mentors.
Speaking at a diversity event in Edinburgh, the first minister said: "My ambition for all our young people is that they will have a fair chance to succeed, and if they work hard and have the ability, there's nothing to stop them realising their dreams.
"It is not acceptable in 2017 for women to be under-represented in senior positions, including in the boardroom."
Ms Sturgeon insisted that her government offered "ambitious commitments" in support of women's equality which included;
She explained that one of her first moves as first minister was to appoint a gender-balanced cabinet.
The chief executive of the Young Scot charity, Louise Macdonald, said that supporting young people in realising their career potential and meeting their ambitions was the responsibility of every leader in Scotland.
She added: "Mentoring is a powerful way of helping young people achieve their goals and will help to create a new generation of positive role models to inspire other young women in the future.
"We're delighted to support the first minister and the Scottish government with this new mentoring programme and we are encouraging as many young women from across Scotland as possible to apply for this amazing opportunity."
"I hope that my election as first minister does indeed help to open the gate to greater opportunity for all women.
"I hope that it sends a strong, positive message to girls and young women, indeed to all women, across our land - there should be no limit to your ambition for what you can achieve.
"If you are good enough and if you work hard enough, the sky is the limit and no glass ceiling should ever stop you from achieving your dreams."

Some officers were sent an instruction not to wear the charity badge following the death of PC Dave Phillips on Monday, the federation's chairman said.
PC Phillips died after being struck by a truck in Wallasey, Merseyside.
A Met spokesman said the badge was not on the approved list but supervisors could allow it at their discretion.
The grey Union Jack badge with a thin blue line running through it is produced by the Care of Police Survivors charity.
The Metropolitan Police Commissioner, Sir Bernard Hogan-Howe, said he personally did not have an issue with officers wearing it, the Sunday Express reported.
"The Metropolitan Police Service has a dress-code policy to clarify the dress standard expected from all staff, whether they are wearing uniform or plain clothes," a force spokesman said.
The approved badges that Met officers can wear include the Police Memorial Day badge, the Royal British Legion's Remembrance Day poppy and the Help for Heroes badge or wristband.
The police spokesman said there was some flexibility, however, as supervisors also had it within their power to permit other badges to be worn at their discretion because different causes emerged at different times.
John Tully, chairman of the Metropolitan Police Federation that represents rank and file officers, called for a "common-sense" approach.
He told the BBC he was made aware on Tuesday of "an instruction that was cascaded down from senior officers" reiterating the Met Police's uniform regulations, but he did not think it should be enforced on this occasion.
He said: "I felt the circumstances were such that common sense should allow this badge to be worn. It was in the context of police attending the memorial service for PC Keith Blakelock and having lost another colleague who died in service earlier this week.
"Morale is pretty low at the moment, and this has the dual purpose of raising money for charity and for paying tribute to fallen colleagues."
Mr Tully said it was unclear who had authorised the instructions to be sent out to officers recently and said the federation had not had sight of the exact wording, but posts had appeared on social media from Met officers using pen names complaining they had been instructed not to wear the Union badge.
He said the death of PC Phillips was devastating and had a profound impact on police officers across the country.
"This death is more unusual because he died while directly serving the public. I think that is why there has been a strong reaction to this. The charity has raised more than £140,000 for his family and it's still rising," he said.
The badge sold out online on the charity's website earlier.
Scotland's First Minister Nicola Sturgeon has launched a competition to find a woman aged between 18 and 23 to mentor for a year.
[NEXT_CONCEPT]
Met Police have been asked "not to wear a Union badge" in tribute to fallen colleagues because it breaches uniform policy, the police federation said.