From the Super Mario Wiki
In Mario & Sonic at the Olympic Games, they are used as items in Dream Events, such as Dream Race in the Wii version, where they are referred to as High Speed, and Dream Long Jump in the Nintendo DS version, where they are referred to as Speed Boost. The player can use them to automatically speed up for a set period of time. It is similar to the Mushrooms in the Mario Kart series.
In the DS version of Mario & Sonic at the Olympic Winter Games, they return once again in the Fever Hockey and Rocket Ski Jumping events, where they are called High Speed Shoes. They are used the same way as in the previous game.
In the 3DS version of Mario & Sonic at the Rio 2016 Olympic Games, they act in much the same way as previous games. They appear in the Boxing Plus event, where they are known as Power Sneakers, and as an effect in the Football Plus event, where they affect the whole team and are known as Speed Up items.
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