The world "challenge" looks easy, but in fact, it is difficult for Japanese people to use this word.
Are the following sentences okay?

1) I will challenge the entrance examination. (Can I use this expression, if I want to enter a very difficult university, for example?)

2) I will challenge myself to enter the university.

3) That service station is challenging to become the number one service station in Japan.

4) That service station is challenging itself to become the number on service station in Japan.
Thank you in advance.
You are right that these uses are VERY common in Japanese-English– so much so that I find myself using #1 with students now.

Nevertheless, none of the uses you have presented are usual in AmE. It would be a good idea to look through the examples of the various uses of the verb [url=http://dictionary.reference.com/browse/challenge?r=66 ] HERE[/url] to better understand how it is used.
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Can I say it's challenge me? or it's challenge my skill.
No, but you can say 'It'll challenge me' or 'It'll challenge my skill'.
The use of the "challenge" is not a challenging task!

Challenge is a call that is used to make a person get involved in a task that requires specific skill.

Think of the following sentences:
1) The entrance exam is challenging! I will try to crack it!

2) Entering that University would be a challenge to me.

3) The service station is trying to become #1 in Japan (Whom is the station challenging?).

4) The employees in the service station are challenging themselves to become the best in Japan.

When I get such kind of doubts I try to figure out the exact meaning of the word. I believe that would help you frame the sentence.
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how about these sentences?

It is challenging me

It is challenging my skill.