I noticed that in many sentences ці/це/ця/цей is accepted only as "these" or "this". Even though the tip gives also the option "the". Should "the" be accepted as well or is this similar to the problem with Russian єтот/єта/єто/єти which is used in Duolingo to mark "the" in English?
Demonstrative pronoun цей has the meaning of "this/that". "The" is definitely weaker. The English courses from Ukrainian and Russian use цей/этот as a translation for "the" just because there is no better translation for "the" in those languages, and they have to differentiate between English sentences using "a" and sentences using "the". But it doesn't mean it is a good translation. Many Ukrainian and Russian sentences in those courses sound pretty awkard. In fact, the English course from Polish doesn't do that, resulting in more natural sentences.
In vast majority of cases where English uses "the", Slavic languages use no word. For example, all non-quoted uses of "the" in both your and my comment would be left out in a Ukrainian translation. The vegetables do not cost much" in most contexts translates to "Овочі не коштують дорого".
As a side note, have you noticed many Slavic speakers often omit articles, both definite and indefinite, when writing English?
This sentence makes sense, however I'm confused - if дорогі /дорогo means expensive how can it also mean 'much' as indicated in the answer ? Literal translation might be "these vegetables do not cost expensive" obviously incorrect English grammar; I've noticed literal translations are often not counted as correct.