The opening panels of the story are set in the Duckburg museum, where Scrooge McDuck is opening a museum exhibit featuring the greatest wonders he has collected during his travels around the world (most of them direct tributes to classic Barks stories).
As Scrooge is bragging to his nephews, Donald Duck and Huey, Dewey and Louie (who all familiar with the artifacts, having taken part in the expeditions for them), Flintheart Glomgold, who is about to open his own exhibit, overhears Scrooge and the conversation between the two rivals turns into a bragging match as to who is the greatest adventurer and treasure-seeker. Scrooge challenges Glomgold to think of something he could find; Glomgold is momentarily nonplussed, then catches sight of Scrooge’s exhibit of Inca artifacts and points out that while Scrooge found the original gold mines of the Incas, he never found the golden artifacts that had been extracted from the mines. Soon Scrooge and his nephews are off on a race with Glomgold to see who can find, and claim the "greater Incan treasure".
The first clue comes as soon as Glomgold has left, as Donald picks up an Incan vase that was knocked over during the bragging match. They find a metal plaque baked inside, providing a map to a temple of Manco Capac in the Andes mountains. Unfortunately, Glomgold is eavesdropping.
Arriving at a village near Cuzco, Scrooge hires a plane to fly them to its location. The pilot of the plane turns out to be Glomgold, who relieves them of the plaque at gunpoint and then parachutes out. Scrooge tries to regain control of the plane and, in a comic episode, inadvertently rips out the belly of the plane while flying too low, dumping his nephews onto the valley floor, still in their seats. As the plane flies off, Glomgold approaches and informs the ducks that Scrooge has frightened away the porters he hired, so they will have to do.
A week later, Glomgold and his reluctant helpers reach a remote mountain, on the summit of which is the temple, built around a large volcanic fumarole (hence, the plaque’s description, the "life breath" of Manco Capac). Glomgold enters the temple’s treasure chamber and is beside himself with glee to discover an enormous store of golden Inca artifacts. Then Scrooge appears, calmly informing Glomgold that he crash-landed the plane on the mountaintop several days ago, and has already filed his claim on the gold using the plane’s radio.
It seems the Scrooge has won, but Huey, Dewey, and Louie are confused about one thing: the plaque makes reference to an Incan "treasure" being moved to the temple, but it actually predates the time of the conquistadors, which is naturally when the gold would have been moved there. Realising there must be another Incan treasure in the temple, Glomgold investigates further and discovers the "Eye" of Manco Capac: an enormous, disc-shaped sunburst festooned with enormous gemstones. Since Scrooge claimed the gold, and not the temple, and there’s no gold on the sunburst, that makes it Glomgold’s.
As Scrooge and Glomgold begin to argue about whose treasure is of greater value, Glomgold begins taking it down from its wall mounting, but it falls and rolls down the temple steps and into the fumarole. It wedges into the hole convex side down, creating a perfect seal. As the volcanic gases build up an enormous pressure, Scrooge notices that the back of the sunburst is sheathed in gold, starting another furious argument between him and Glomgold, and causing them to wedge the sunburst down even more firmly. Before the others can stop them, the pressure mounts and the entire mountaintop, temple and all, is suddenly blown into the sky like a cork from a bottle. The ducks are able to use a tapestry as a makeshift parachute before the temple lands squarely in a nearly bottomless volcanic lake, next to the village they originally started from. The massive splash of water irrigates the villagers’ crop fields, relieving them from the effects of recent drought.
All of the treasure is now completely irretrievable. As the dispirited ducks begin their journey back to civilisation, Scrooge is seen emerging from the village and mentions that he has agreed to build a pumping station for the village so that they will never be troubled by drought again. Glomgold scorns Scrooge's generosity, until Scrooge reveals that in return, the villagers have agreed to sell him the lake for one peso — which makes Scrooge the rightful owner of both the temple and all of the treasure inside it. Even though he cannot retrieve it, Scrooge is now the clear legal owner of all of the treasure — and thus the winner of the contest.

Question: Who filed the claim on the gold using the plane's radio?
Answer: Scrooge
Alternative answer: Scrooge