1) I will challenge the entrance examination. (Can I use this expression, if I want to enter a very difficult university, for example?)
No, this is not a normally-heard use of the word "challenge."
- "I will ATTEMPT the entrance examination" expresses the meaning, even if a little more weakly than "to challenge" usually suggests.
- Lately in Academia the expression has grown up: "to challenge a course" or some similar variation. This expression means that the student will take an exam that will allow him to skip a required course if he passes the test, to earn credit for a course by an exam similar to a placement exam.
- In student life, the expression is "I tested out of Math 101" or "I'm going to try to test out of Algebra I."
- Here is a link to a university's description of its procedure for challenging a course
Challenging a Course - Abilene Christian University
- However, this usage does not apply to the phrase in the question.
2) I will challenge myself to enter the university.
Yes, this is all right -- but barely so.
- "To challenge oneself" really refers to actions and behavior that are solely within the power of the self. But "challenging oneself to enter the university" is not really a situation that is up to the student alone -- the university also has some say in the matter.
- The reader has to add his own interpretation to the phrase, and to assume it means that the student has vowed to persevere until he succeeds, or that he will do his level best on the entrance exam, or that he will knock himself out prepping for the entrance exam.
- Perhaps a better phrase would be
~ "I am committed to getting in to that university" or
~ "I will do whatever it takes to enter the university" or
~ "I have dared myself / promised myself that I will enter the university -- come what may."
3) That service station is challenging to become the number one service station in Japan.
No, this is not a possible usage.
- "The service station is STRIVING to become the number one service station in Japan" is a perfect rendering of the meaning, including the implication of effort and struggle.
- There are contexts in which it is okay to say
~ "The District Manager has challenged this service station to become the number one service station in Japan." or
~ "This service station has been challenged (by some outside agent) to become the number one service station in Japan."
- But that is not the meaning of the sentence in the question.
4) That service station is challenging itself to become the number one service station in Japan.
Yes, this is perfectly fine.