Ms Flanagan Willanski said she "just want to go on record as having asked you to take climate change seriously" but claims Mr Abbott responded by offering to bet her $100 that the climate would not change in a decade.
A picture of a piece of paper with Mr Abbott's signature accompanied the article.
"Afterwards I donated $100 to the Australian Youth Climate Coalition. I gave away my winnings 10 years in advance because I knew I’d won this ridiculous, offensive bet as soon as it was made," Ms Flanagan Willanski wrote.
On social media, voters have responded in disbelief that Mr Abbott would make the bet with a complete stranger.
"Only the hardest core denier would make this bet. Tony Abbott is certain to lose," University of Sydney professor Tim Stephens tweeted.
"The level of contempt this shows Abbott to have for the rest of us is spectacular," wrote another user.
The revelation comes just days after the Liberal member for Warringah was confronted by two 15-year-old local residents and grilled on climate change.
"For me, and especially for all of my peers, we just think that climate change is such a big issue and we just want you, as my representative because I live in Warringah, to understand that we want climate action now," Elsie Loadman told Mr Abbott on Friday.
But Mr Abbott responded that the financial cost of climate action was high and added that it was his duty, as their local member, to "tell them to truth not just agree" with the students.
"He was just denying all the scientific facts," Yasmin Lin said.
"I was so mad that I couldn't speak for a period of time."
Earlier in the day, hundreds of students had gathered outside his Manly office to protest what they see as a lack of government action on climate change.
Mr Abbott has long been viewed as a climate change sceptic, telling reporters during his campaign launch last month that the "so-called settled science is not quite as settled as people say".
The issue has become a key battleground in Warringah, as Mr Abbott goes up against Independent Zali Steggal to win the seat he has held since 1994.
The need for climate action has been seized upon by former Olympian Ms Steggal as her main policy platform.
Mr Abbott's office has been contacted for comment.
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