Spirit of Youth – the Singapore 2010 Youth Olympic Games logo – celebrates Singapore 2010 in an adventurous, open and fun way. It evokes the Singaporean sense of confidence and progress. A red flame represents the passion to learn and the power of positive thinking. A purple star symbolises excellence and the pride of representing one’s nation. A crescent reflects dynamic youth, full of promise. The active appearance of the “Singapore 2010” word mark comprises individual and unique characters that signify harmony among the diverse participants. The Spirit of Youth inspires all of us to express the Olympic spirit and celebrate the first-ever Youth Olympic Games.
Cite: http://www.sgyoutholympics.com/2009/01/13/singapore-unveils-official-logo-of-the-inaugural-youth-olympic-games/

These five rings -- blue, yellow, black, green and red -- represent the five parts of the world now encompassed by Olympism and ready to compete against each other. Moreover, the six colors (including the white background) thus combined represent those of all nations, without exception. The blue and yellow of Sweden, the blue and white of Greece, the French, English, American, German, Belgian, Italian and Hungarian tricolors, the yellow and red of Spain are side by side with the new Brazilian and Australian flags, the old Japan and the new China. It is a true international emblem.

The Logo of the Beijing 2008 Olympic Torch Relay owes much of its inspiration to the traditional concept of "fire phoenix", and presents the image of two runners holding the Olympic Flame high. As ancient Chinese legend has it, phoenix is the king of all birds, and symbolizes good fortune, eternity, nobility and happiness. The use of the phoenix image in the Torch Relay logo conveys the idea that the Torch Relay will send the best wishes from the Beijing Olympic Games to people all over China and the rest of the world.
Done by: NeoHuiLing.(12) Pictures could not be uploaded. I have post on 7/5 but i came to edit.