The top delegates from Pyongyang delivered their leader's letter to President Moon before having lunch together.
Moon, however, dismissed the proposal, saying it is a matter of his country's sovereignty, the office added.
Among the crowd was a large number of North Korea's all-female "Army of Beauties" cheer squad, known for their tightly choreographed moves and chants.
The ceremony was held at the PyeongChang Olympic Stadium in South Korea's eastern county of PyeongChang.
Pence said Friday the US would oppose talks between the two Koreas until the North agreed to open negotiations on ending its nuclear program.
Warmbier said that even though North Korea had formed a "unified" team with South Korea, its athletes were not truly demonstrating Olympic spirit.
"They are really attractive", added Biney, who qualified for quarterfinals in the 500 by placing second in her heat.
To fill out their team their hockey officials brought in American and Canadian players who had at least one Korean parent.
Pence's tweeted day-after position: "The U.S. will not allow the propaganda charade by the North Korean regime to go unchallenged on the world stage". Moon and Vice President Mike Pence watched the proceedings from the VIP balcony in the stadium, and they were joined by Kim Yo Jong. The North and the United States were also holding a series of high-level talks on Pyongyang's nuclear and missile programmes.
But the North was welcomed with open arms to what South Korean President Moon Jae-in called "Olympic games of peace". Moon, a dove who wants to resolve the nuclear standoff diplomatically and peacefully, quickly responded to Kim's outreach by offering talks.
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I hope people are now seeing & understanding what is going on here. "DRAIN THE SWAMP!" the president tweeted. That, at least, is the official story as reported by The New York Times Friday night.
The spokesman says Moon replied that the North and South should continue to work to build conditions so that a summit can take place.
Pence, who is using his visit to South Korea to underscore the Trump administration's policy of maximum pressure on the North, has publicly and repeatedly warned Seoul not to let down its guard to a North Korean charm offensive.
"As an athlete myself it was a great honour for me to be the last torch bearer", she said.
Pence and Japanese Prime Minister Shinzo Abe sat nearby, looking expressionless.
Regardless of obstacles, Saturday's meeting marks a historic milestone in inter-Korean relations.
Cheerleaders of North Korea hold unification flags during the opening ceremony at the Pyeongchang Olympic Stadium on February 9. It has dispatched a delegation of almost 500 people - mostly musicians, dancers, and an all-female cheering squad - and has been pushing its participation as a sign of willingness to work with Seoul, through greater exchanges, to ease what has been a year of very high tensions on the peninsula.
The officials toasted with the Korean liquor "soju" and were served buckwheat and potato pancakes, dried pollock soup, grilled halibut and other dishes.
A few months before the competition began, Korean Air Flight 858 was downed in a bombing later attributed to North Korean agents.
This item corrects that Kim Yo Jong is the leader's sister.
The invitation, delivered by Kim's visiting sister Kim Yo Jong, said Kim was willing to meet the South's leader "at the earliest date possible", said a spokesman for the presidential Blue House.