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Although not firmly established in canon, the latest movie giveds two vital clues, Spock birth in 2230, and Spock already being an instructor by 2254 when he programs the Kobayashi Maru. Allowing for the generally accepted 4 year program, and two years of intructing (Kirk also taught Mitchell at the end or right after the Academy as a Lieutenant) we can suppose Spock entered the academy at 18, in 2248.

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