From the Super Mario Wiki
Synchronized Swimming is a Aquatics Event that appears in the Wii version of Mario & Sonic at the London 2012 Olympic Games. The objective of the event is to score the most points, with the event taking place in the Aquatics Centre. Amy, Peach, Daisy, and Blaze can be seen competing in this event in the game's opening. The starting Olympic record in this event is held by Team Amy at 46.58 points.
After all players have chosen a song, the events start for the first player. The songs that can play are Spring Song, Trisch-Tracth-Polka, and Hungarian Dance NO.5, in order of ascending difficulty. This events starts with the player's team being announced, before showing the starting platform. Here, the player is taught how to pose. Then the player must swing the Wii Remote down or wait for it to start. Points are earned for both if done. Players need to watch the pictures that appear onscreen, and copy them with the Wii Remote. Sometimes, there are doubles that the player need to act fast on. If there's a mark on a line, the Wii Remote has to be swung down, but sometimes up. There are also shapes that the players Swing the Wii Remote down multiple times. If there's a spin, the Wii Remote has to be spun around. At the end of the song, the player needs to pose with the Wii Remote. If multiple teams are playing, it goes to the next team.
The Synchro Teammate challenge is completed by getting a chain of 30 moves a number of times in this event. The criteria for unlocking each achievement are as follows: