Competing

go for the gold
成功のために全力を尽くす (less translation, more explanation...)

Comment:
This probably originates from the Olympics, where getting gold means winning, but it is used generally to mean trying hard to win.

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If you can't beat them, join them.
長いものには巻かれろ。

Comment:
No point trying to win something you can't win.

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give it one's all
全力を尽くす

Example:
Obama gave it his all to win the Democratic nomination.
オバマは民主党の大統領候補指名のため全力を尽くした。

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sweet victory
胸のすく勝利

Example:
Winning that long match was a very sweet victory.
あんな長い試合を勝ったのは胸のすく勝利やったな。