Whenever a permanent you control enters tapped, untap it.
"Everyone is obsessed with this cosmium stuff, but I can sense some kind of comparable power in this droplet of ancient amber." —Quintorius Kand, research notes
2010-03-01
For Amulet of Vigor's ability to trigger, a permanent must enter tapped due to an effect that says "put [the permanent] onto the battlefield tapped," "[this permanent] enters tapped," or the like. If it enters untapped, the ability won't trigger, even if you tap that permanent afterward.
2010-03-01
If you control more than one Amulet of Vigor, each Amulet's ability triggers when a permanent enters tapped and under your control. The first ability that resolves will untap that permanent. If the permanent somehow becomes tapped again before the next ability resolves, the next ability will untap it as well (and so on).