The climax of "The Most Dangerous Game" comes when Rainsford jumps into the ocean. 

Zaroff has been chasing him for days. After Rainsford failure on the first day, Zaroff played "cat and mouse" with Rainsford, letting the prey remain free for a while. When Zaroff comes again for Rainsford, Rainsford begins to use his wits and his experience, killing off one of Zaroff's dogs and later Zaroff's assistant, Ivan.

Despite his successes, Rainsford becomes cornered in a section of the island. The dogs are coming for him and he has nowhere left to run. The tension of the story reaches its peak as Rainsford makes a decision about how to respond to this impossible situation. 

In a fit of desperation, Rainsford looks to his only escape—jumping off the cliff into the sea which waits far below. He takes this chance.

After this moment in the story, we are presented with conclusion and resolution. Though Zaroff and Rainsford have their final confrontation in the house and Rainsford kills Zaroff, this final meeting occurs by way of falling action.

The tension of the story has already begun to dissipate. Others might argue that the concluding confrontation between the two men represents the climax of the story, bringing the central conflict to a close. However, the central conflict of the story - Rainsford's battle for survival - is decided when Rainsford jumps into the sea. 

This question is open to debate. In my opinion, the climax to "The Most Dangerous Game" comes near the end of the story when Rainsford finds himself trapped and is forced to make a decision: whether to stand up to the approaching Zaroff or to take his chances with the dangerous waters below.

     "Nerve, nerve, nerve!" he panted, as he dashed along. A blue gap showed between the trees dead ahead. Ever nearer drew the hounds... Twenty feet below him the sea rumbled and hissed. Rainsford hesitated. He heard the hounds. Then he leaped far out into the sea...

However, author Richard Connell imposes a double-surprise ending which also presents the possibility of an alternate or different climax. Not only does Rainsford survive the plunge into the sea and then surprise Zaroff in his bedroom, but he then announces that the hunt is still on, this time with Zaroff as Rainsford's prey. This could also be considered the climax, but I would call it the falling action. The last line of the short story serves as the denouement, the final resolution.

     He had never slept in a better bed, Rainsford decided.

For this story, you need to examine the events carefully to determine the main turning point, which is a good indicator of the climax. Where in the story does the action or a character change?

I think the most obvious answer is near the end of the story when Rainsford confronts Zaroff in his bedroom. Zaroff tells Rainsford that he is free to go, since he won the game. At this point Rainsford has the opportunity to show his moral superiority to Zaroff, but instead, he chooses to continue the game.

Interestingly enough, after having killed Zaroff, Rainsford does NOT free the "prey" that Zarroff has been holding for future hunts. This inaction further supports the idea that Rainsford's confrontation with Zaroff is the climax of the story, since the possibility of his replacing Zaroff (symbolized by a comfortable night's sleep in Zaroff's bed) definitely indicates a change, or turning point, that takes place in Rainsford's character.

Throughout the story the audience has viewed Rainsford as different from Zaroff, but at the climax his true--or possibly his new character is revealed.

The climax of a story is the point in the narrative when the intensity and tension are the highest. In Richard Connell's classic short story "The Most Dangerous Game," the climax takes place on Rainsford's third day on the island when he is forced to jump into the ocean to avoid the maniacal general and his pack of hunting dogs.

On the third day, Rainford sees General Zaroff walking alongside Ivan, who is being led by a pack of hunting dogs. Rainsford knows that he must act quickly and fashions a booby trap that he learned to make in Uganda by tying his knife to a young sapling, which is held back by some grapevine. Rainsford's trap works and ends up killing Ivan but does not halt the general's advance. The intensity of the story increases to a fever pitch as the general and his pack of dogs close in on Rainsford, who has run out of options and is in a precarious situation. Rainsford can hear the dogs barking and knows that Zaroff is close by. Rainsford makes the daring decision to sprint towards a cliff and leap into the ocean to avoid the general and his pack of hunting dogs.

The climax of Richard Connell's short story, "The Most Dangerous Game," comes when Sanger Rainsford manages to elude the murderous Russian Cossack, General Zaroff, diving off a cliff into the waters below in his last-ditch effort to escape his hunter. Rainsford had managed to elude Zaroff for more than a day, killing Zaroff's companion, Ivan, and one of the dogs; Zaroff, too, had been injured. But Rainsford realized his time was running out and, when reaching the cliff, he decided to take his chance with the dangerous dive rather than with Zaroff's relentless pursuit. The latter paragraphs of the story, when Rainsford returns to Zaroff's bedroom, serves as more of a falling action after the exciting hunt.