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The phrase "a new problem exists" is correct and usable in written English.
You can use it when stating that there is a newly identified issue or challenge that needs to be addressed.
Example: "After reviewing the project, it became clear that a new problem exists that we need to tackle immediately."
Alternatives: "a new issue has arisen" or "there is a new challenge".
Exact(2)
A new problem exists.
A few decades later, a new problem exists when students and engineers do not really see the relationship between basic knowledge and real engineering problems.
Similar(58)
Decontamination of buildings from intentional release of B. anthracis is a new problem, and no accumulated scientific knowledge exists on the subject.
Then a new problem arose.
And now they have a new problem.
Every job is a new problem.
It's not a new problem.
It is not a new problem.
It's not a new problem either.
It was not a new problem, though.
They also introduced a new problem, corrosion.
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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