What is the Korean for "to throw one's hat into the ring"?

ANSWER: 출사표를 던지다

When one "throws one's hat into the ring," it means the person wants to challenge someone in a contest or something. Today the expression is often used in politics to mean a political candidate wants to challenge the current officeholder, but originally it was used to challenge someone to a boxing match. In other words, someone outside a boxing ring could throw his hat into the ring to challenge the champion there.

In the Korean expression 출사표를 던지다, the verb "던지다" means "to throw," but 출사표 (出師表) does not mean "hat."

The phrase 출사 (出師) means "to dispatch troops." The Chinese character 出 (출) means "to send out" or "to dispatch," and 師 (사) means "troops," though today 師 is more commonly used to mean "teacher" or "master" (스승).  The 표(表) in the expression means "memorial" (장표 章表), which were written documents the Chinese used to communicate with their emperors. So, 출사표 (出師表) literally means "a memorial to dispatch troops."

One of the problems with using 출사표를 던지다 to mean "throw one's hat in their ring" is that the Chinese would have never "thrown" their memorials at their emperors, at least none who wanted to live.

By the way, there is a really cool Korean song called "출사표" that starts at about the 2:50 mark in the following first video. The first video includes the Korean lyrics with English translations. The second video is a live performance, which is also cool.

The song seems to suggest that a young person has decided to finally leave home to take on the challenges of the world. In other words, the person has decided to throw his or her hat into the ring.