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The phrase "principal problems" is correct and can be used in written English
It is typically used to refer to the most significant or important issues in a situation or problem. Example: One of the principal problems facing our company is the lack of diversity in our hiring process.
Exact(37)
Four years on, the principal problems remain.
The government had two principal problems: to make peace and to restore peacetime finance.
For bondholders, the carrot is a greater prospect of getting their money back.There seem to be three principal problems.
For the historical development of the different sciences and their scope, component disciplines, methods, and principal problems, see physical science; Earth sciences; biology; medicine, history of; engineering; social science.
When the Italian government declared a state of emergency for Pompeii in 2008, Culture Minister Sandro Bondi cited the strays as among the principal problems, along with illegal tour guides, inadequate washroom facilities and general neglect of the site.
Politicians are betraying their commitment to serve the public interest it they cannot accept the challenge to reframe the political debate around an understanding that the principal problems we all face are global and can only be tackled by action at the global level.
Similar(21)
I call that the "dumb principal" problem.
That's the other principal problem.
"The principal problem is our schools.
Therein, perhaps, lies its principal problem.
Ashe's principal problem in tennis has been consistency.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com