A Goryeo dynasty (AD 918-1392) bronze bottle from the Korean collection of the Smithsonian Institution, and not in fact the Korean FA Cup.
The 2012 Korean FA Cup began in March with 8 clubs from the third-tier Challengers League, 7 from university competitions, and 1 from the reserve league. In April the 8 winners were joined by the Challengers League champion plus 5 clubs from the second-tier National League. In May the 7 winners were joined by the top 9 from the National League and all 16 K-League clubs in the Round of 32. June 20th was the Round of 16. Today it's the quarterfinals.
Are Pohang Steelers still in it? Yes. They entered at the Round of 32 and beat Cheongju Jikji 4 - 0, then in the Round of 16 they beat Gwangju FC 3 - 1. Today they play Jeonbuk Hyundai Motors.
Pretty much the first notable event of the London Olympics was the Korean flags screw-up, in which pictures of North Korean players were displayed next to a South Korean flag. Ouch! So, to remove any ambiguity in this article, let me just state that the Korean FA Cup is contested in South Korea. North Korea has the Republican Championship, which could stand some better publicity. Though you can get your North Korean football hit here.
[Pohang win 3 - 2! Incidentally, Steelers' No Byung-Jun was Man of the Round in the Round of 16.]