What do the boys determine at the top of the mountain?
There are a few possible answers to your question depending on when in the book you are referring to.
“It was furry. There was something moving behind its head—wings.
The beast moved too—”
“That was awful. It kind of sat up—”
“The fire was bright—”
“We’d just made it up—”
“—more sticks on—”
“There were eyes—”
This is Samneric telling Ralph and Piggy and the rest of the boys what they have seen at the top of the mountain when they are supposed to be watching the fire: they think they have seen the beast. What they have actually seen is the silhouette of the parachutist who has landed on the top of the mountain - though what they think they have determined is that the beast is real, and that there really is something to be scared of.
Then, when they take a mission up the island to look for the beast, Jack discovers the great place for a "fort", which will later become "Castle Rock". And then Simon decides to go up the mountain on his own, and goes running off before anyone can argue.
What he determines at the top of the mountain is the truth about the parachutist and the beast:
Simon felt his knees smack the rock. He crawled forward and soon he understood. The tangle of lines showed him the mechanics of this parody; he examined the white nasal bones, the teeth, the colors of corruption. He saw how pitilessly the layers of rubber and canvas held together the poor body that should be rotting away.
The beast was harmless and horrible; and the news must reach the others as soon as possible.
But - of course - it never does: and Simon is killed before he can deliver his news.
Hope it helps!