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"exist problems" is a correct and usable phrase in written English.
You can use it to describe an existing set of issues, such as in the sentence: "Although there are many benefits of this policy, it unfortunately also exist problems that need to be addressed".
Exact(7)
On the "last resort": "There do not exist problems that cannot be solved around a table," Levi wrote, though he added, crucially, "provided there is good will and reciprocal trust".
There exist problems in applications whose models can be written in the form [9]. (1.4).
Ivanovs and Palmowski [2] discussed the exist problems for spectrally negative SN-MAPs, and gave some identities for two-sided reflection SN-MAPs.
There exist problems for which the most efficient known decision algorithm has exponential time complexity in the worst case (and in fact are known to be \ \textbf{NP}\ -hard in the general case – see Section 3.2) but which operate in polynomial time either in the average case or for a large subclass of problem instances of practical interest.
There exist problems in applications whose models can be written in the form [2] x ′ ( t ) − q ( t ) x ′ ( g ( t ) ) + ∑ i = 1 m b i ( t ) x ( h i ( t ) ) = f ( t ), t ∈ [ 0, ∞ ). (4). This equation is a particular case of (1).
end{aligned} (12) Also, when X has paths of unbounded variation, the proof of exist problems often relies on an approximation idea, which has already appeared in various papers; see Albrecher and Ivanovs [3], Czarna [9] and Baurdoux et al. [10].
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This paper analyzes the common reliability design methods and the existed problems.
On the other hand, where true religious and social intermingling exists, problems are few.
The present study shows that there exist problems of inequality in health among older adults in the study setting.
Actually, there exist subtle problems with such a construction.
Hence, there exist interoperability problems among them.
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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