I have looked again,and "A" is the right answer. But my teacher say, "A.B.D can all be the answer."

Actually, it is "A". "Proved" works as a participle here (the past participle of "prove" can be either "proved" or "proven").

"A" is correct. This is very advanced English, and you will have to learn these structures. You can't just second guess your way through an advanced grammar test.

This one is difficult. With her efforts to build team spirit proved fruitless = With her efforts to build a team having been proved fruitless = Once her efforts to build team spirit were proved fruitless. I would have stated it the second way. You can see that the phrasing in the question is confusing. You see that construct "with + <subject> + + <adjective> " in written English, but almost never spoken.

Look again: 'A' can't be the right answer. Maybe you're reading the answers to another exercise?