Through the first week of the Olympic Games in Rio, it has been moving to see so much of what has lately been generating anxieties at home contribute so powerfully to American pre-eminence on the international stage. If race, gender, immigration and even our definitions of success are dividing us as citizens and voters, they are uniting us, if only temporarily, as supporters of Team USA.

In the women’s gymnastics competition, our national strength comes from diversity. World champion Simone Biles, gold medal-winner in the individual and team all-around competitions, is African-American, as is Gabby Douglas, who won gold medals in those events at London 2012. Laurie Hernandez is Puerto Rican, and the team is completed by two white gymnasts — Aly Raisman, who is Jewish, and Madison Kocian. This line-up stands out in a political season animated by a dark winner-takes-all mentality suggesting that success for one group of Americans can come only at the expense of another.

The irony is that what makes for a divisive falsehood in an election is, in fact, true of Olympic gymnastics. Only five women could make the US team. Rules preventing one country from crowding out others mean that only two US women could reach the finals for the individual events; Hernandez, for example, would be competing in the floor exercise final tomorrow had Biles and Raisman not qualified with even higher scores. If Biles were not so broadly dominant, and if women’s coach Martha Karolyi were not known for putting together Olympic teams with a strong balance of skills, it might be easy for viewers to fragment into partisan rooting factions, cheering for one gymnast over the other.

Instead, their accomplishments feel collective, their individual excellence lifting the team as a whole. And, in a small way, their victories belong to all of us, putting the lie to the idea that we cannot succeed together.

Over at the aquatics stadium, a different facet of US achievement has been front and centre. Swimmers from a number of countries spoke out against competitors caught using performance-enhancing drugs in the past, with American Lilly King’s condemnation of the Russian Yulia Efimova attracting particular attention. Episodes of wagged fingers and splashed water have been read as the latest and wettest — if not the coldest — chapter in the fraught relationship between these two nations. But King’s gestures resonate for a different reason. At a moment when the authenticity of Republican Donald Trump’s business success has become an important issue in the presidential election, it is heartening to see Americans abroad argue that when we win, we want it to be real — even if that means disavowing accomplishments we come by dishonestly, as King has done in condemning Americans who are competing in Rio despite past failed drug tests.

Trump may be able to conceal his true wealth by refusing to release his tax returns, and to hire foreign workers to staff his properties even as he promises to bring jobs back to America, but in the Rio pool you cannot rely on prestidigitation to claim the gold. Similarly, as Trump’s campaign continues to smash invaluable international and political norms, whether he is pondering the use of nuclear weapons, the assassination of his opponent or an amped-up use of torture, there is power in seeing an American declare that we define ourselves in part by what we don’t do.

By now in this column, you have probably noticed another theme of the Rio Olympics: the prominence of American women. Whether it is swimmer Katie Ledecky’s and Biles’s supremacy, or King’s emergence as a moral voice in the doping controversy, women have given American audiences some of their strongest occasions to cheer.

Also noteworthy are the contributions that immigrants, including Australian-born basketball player Kyrie Irving, who in 2012 decided not to play for his country of birth so he would be eligible to compete for a spot on Team USA, are making to the United States medal count. In a political context, reactions to America’s increasing diversity, the prospect of female leadership, the presence of immigrants and contested ideas about what constitutes genuine success have congealed into a moment of extraordinary ugliness. Yet when we measure ourselves against other nations, we see the beauty and strength in inclusivity and integrity.

Speaking on the campaign trail last Thursday, Hillary Clinton used Team USA’s success to blast Trump, suggesting that if American athletes feared the outside world as much as her competitor did, “Michael Phelps and Simone Biles would be cowering in the locker room”.

She could have made a simpler point. As Rio is making unmistakably clear, the parts of American life that Trump wants to change to “make America great again” are some of the very things that make America great just as it is.

• Alyssa Rosenberg blogs about pop culture for The Washington Post’s Opinions section