"Over the past few years, my husband and I have seen Americans unite in so many ways across the country," Obama, 51, said.
"These games are a perfect reflection of that unity," she continued. "They show us that we're all in this together – that we can lift up our friends and neighbors, and that we can bring out the best in each other to reach even higher heights.
"We see it in the athletes – the amazing athletes – who give it their all, who lend a hand when somebody falls, who high-five and hug every one of their competitors. We see it in the fans, in the families, the volunteers and coaches who are here to shower our athletes with love and support every step of the way."
Joining the First Lady in the audience was Caroline Kennedy, whose family helped found the Special Olympics in 1968.
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"To all of our athletes, I just want you to know that all those people won't just be cheering for you, they're going to be learning from you," she said. "They're going to be inspired by you. And they'll be reminded that at our core, we all want the same things – a sense of pride in who we are, and a belief that we can reach our dreams no matter what challenges we face.
"That's what you all are doing here. That's your mission. You're bringing us together. You're filling us with so much hope. And you're uniting us in a way that nothing else can."
Obama ended her speech with eight "magical words": "Let the 2015 Special Olympics World Games begin!"