I was looking for an equivalent for a type of school that we have in our country (they usually have entry exams) and I faced this phrase "public exemplary schools". Do we have such schools by this name generally? Is it common?
I am not aware of any well-known term in US English for a school that has an entrance exam. "Magnet schools" are public schools that serve more advanced students, but the name does not indicate how they decide who to admit. "Exclusive schools" are difficult to get into, but again, whether they select based on an entrance exam, grades in previous schooling, or whether your parents are willing to pay high tuition isn't implied by the name.
In the UK, free state-run schools that require students to pass an exam to enter are called grammar schools. They are relatively rare now, only a few regions of the country still have them, but they have been widespread.
The name is because, traditionally, only at a grammar school could you learn Latin Grammar.
I believe that the term is not used in American English. In American English a "grammar school" is a school for younger pupils aged about 6-12. The terms used in American English are quite different to British English; for example in British English, a "public school" is a school that charges fees and is independent of the government. In America, a public school is state-run and doesn't charge.
What this means is, unfortunately, there is no term that is widely understood across all dialects of English for a "public school with an entrance exam"
There is a term that is used in American English for free, state-run schools that require an entrance exam, but it isn't common or well-known, primarily because this kind of school isn't common in the US. We call them exam schools or occasionally competitive public schools (see, e.g., the definition in The Black Student's Guide to High School Success or this Salon article on the subject).
However, so far as I know the exam is generally peculiar to each individual school—that is, schools have their own tests, and you either get in to that school or you don't; there isn't a tiered system where highest marks send you to one school, and lower (but still high) marks send you to a different school. Even if there were such a scheme in some particular school system, there probably still isn't a widespread name to distinguish the two types of exam schools.
In Australia, those types of schools can be called 'Selective schools', as seen in the article below: