China/USSR
Sino-Soviet split
When Mao found out that Kruschev (new leader of the Soviets) had the ability to use ICBMs on the US, he urged them to declaer war on the states. Kruschev decided not to, he wanted to develop a peaceful co-existence with the US. This wasn't good enough for Mao and so he cut off all dealings with the USSR.
Related: Sputnik

Cuba
Fidel Castro
Fidel Castro is the leader of communist Cuba. Cuba became communist during the cold war (not a good time for a country so close to the USA to become communist) and was dealing heavily with the USSR. As the nuclear arms race continued between the US and the USSR, the US was naturally shocked to hear that the USSR was planning to send ICBMs to missile bases in Cuba (now the missiles could reach the states in less than ten minutes!). What resulted was the Cuban missile crises, when the US and the USSR came to nuclear brinkmanship (the closest one can get to war without going to war). Catastrophe was avoided when the leaders of the two super powers met and both decided to pull missile bases out of near each other's territories.
Related: SALT II | Cuban Missile crises | Bay of Pigs | Sputnik | Warsaw Pact, CENTO | SEATO | McCarthism | Berlin Blockade, Marshall Plan | NATO | OAS, Truman Doctrine | Berlin wall

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