A combination photo shows President Donald Trump (L) in New York, September 21, 2017 and North Korean leader Kim Jong Un in an undated photo released by North Korea's Korean Central News Agency (KCNA) in Pyongyang, September 4, 2017.
"I will surely and definitely tame the mentally deranged USA dotard with fire", Kim said in September, after the U.S. president, in his maiden address at the United Nations, called Kim a "rocket man".
Mr Trump has referred to Mr Kim as a "madman" and "Rocket Man" while the 33-year-old responded by calling the 71-year-old former reality TV star a "mentally deranged dotard".
The exercises come on the heels of Trump's visits to Seoul and Beijing this week, which were dominated by the question of how to counter Pyongyang's nuclear weapons threat.
"Why would Kim Jon-un insult me by calling me "old", when I would never call him 'short and fat?' Oh well, I try so hard to be his friend - and maybe someday that will happen!" wrote the president in a tweet on Sunday.
"That might be a unusual thing to happen but it's certainly a possibility", he told reporters at a press conference in the Vietnamese capital.
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But in its retort yesterday, North Korea said Mr Trump's warnings "can never frighten us or put a stop to our advance", but rather "pushes us to speed up the efforts to accomplish the great cause of completing the state nuclear force". "If it did happen, it would be a good thing for North Korea, and a good thing for lots of other places and good for the world". "I don't know that it will, but it would be very, very nice if it did".
White House chief of staff John Kelly, chatting briefly with reporters after Trump's press conference Sunday in Hanoi, was asked about the president's descriptive tweet about the North Korean leader. There always playing politics - bad for our country.
During the very same day Trump said he was prepared to mediate between claimants to the South China Sea, which include Vietnam and China.
"Because of North Korea's nuclear and missiles threat and growing threat, we need to be sure we have a robust military preparedness for whatever contingency might arise", she said.
But Xi said only that dialogue was crucial to resolve the problem. "But it is certainly a possibility", Trump said.
"The giant gift pack of US$250 billion in trade deals is already something Trump can take home and claim as a success of the visit", he said. "The president wants to build upon that to really build our relations with Asean".