Combine the orange juice, grapefruit juice, lemon juice, lime juice, and vodka, if using, in a pitcher. Dip the rim of 6 martini glasses first in a dish with lemon juice and then a dish with sugar. Set aside to dry.
When ready to serve, place ice cubes in a cocktail shaker, add the cocktail mixture to fill the shaker three-quarters full, and shake for about 30 seconds. It's important to shake for a full 30 seconds to get the drink very cold and dilute it slightly with the ice. Pour into the sugared martini glasses and garnish with a sprig of mint. Continue filling the shaker with ice and cocktail mixture until all the drinks are poured. Serve ice cold.
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