ANSWERS: 2
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Here are some steps you can try first: Make sure the Nintendo GameCube has good ventilation. For best results, make sure that the system rests on an open, flat surface. Avoid operating the Nintendo GameCube while it's on carpeting or inside an enclosed space. Check the vents on the sides of the Nintendo GameCube for a build-up of dust. The system will shut down if it cannot vent properly to prevent overheating. If there is a build-up, remove it using a vaccum cleaner with a soft brush attachment. But according to Nintendo: "If the power light flashes once when power button is pressed, then turns off and won't turn back on, the system needs repair." http://www.nintendo.com/consumer/systems/nintendogamecube/trouble_power.jsp
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You may have a damaged memory card or controller. Some non-Nintendo controllers tend to go bad quickly and when a controller is bad, it brings down the whole system (the Gamecube won't function with them attached). Some fixes are to (this is also listed in the instruction booklet) disconnect all controllers, memory cards and other devices (such as a GB player or a network adapter) and try turning it on again. If it works, connect one device at a time, turning the 'cube off between each one, until it stops working. When it refuses to turn on, disconnect the last device and toss it. If it won't turn on even without any devices, it needs to be serviced by Nintendo.
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