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The phrase "a problem exist" is not correct in standard written English.
The correct form should be "a problem exists" to agree in number with the singular noun "problem."
Example: "There is a problem that exists with the current software update."
Alternatives: "a problem is present" or "a problem occurs."
Exact(3)
We are constantly asking ourselves- why does a problem exist?
Only with a possible body dysmorphia diagnosis might a problem exist, but, asking a client if others invariably perceive the disturbing imperfections they are obsessed with can help in assessing whether they suffer from this disorder.
Briggs [ 31], on the other hand has differentiated between i) diagnostic assessment (does a problem exist, is policy action needed?), ii) prognostic assessment (implications of potential policy options, which option to choose?), and iii) summative assessments (effectiveness of existing policies).
Similar(57)
Clearly, a problem exists.
A problem existed though.
But, also unlike ours, theirs admit that a problem exists.
Some Pakistani officers denied a problem existed at all.
Just because a problem exists doesn't mean there has to be a solution to it.
For much of the past 120 years, however, there was widespread denial that a problem existed.
After years of denial, the president now concedes that a problem exists.
A problem exists when there is a discrepancy between the perceived reality and the valued goal.
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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