A/N: I do not own the Potterverse, J.K. Rowling does.
A/N 2: To facilitate the narrative of these stories, as I am basically rewriting bits of canon, I have made note of where these stories would fit in to the books. Italicized text is from the series; the rest is mine.
From Harry Potter and the Goblet of Fire
"Off you go, then!" said Bagman brightly to the four patrollers.
"Good luck. Harry," Hagrid whispered, and the four of them walked away in different directions, to station themselves around the maze.
As Bagman was about to magically magnify his voice to begin the competition, Harry realized he didn't know how the champion who won the Tournament was supposed to exit the maze. Bagman hadn't given them any more information than the first to touch the cup would be declared the winner.
"Mr. Bagman, wait!" Harry exclaimed, surprising the man and the other champions.
"Yes, Harry?" questioned Bagman. "Do you have a last minute query?"
"How does the winner get out of the maze once they have touched the trophy?" asked Harry. "Are we supposed to send up sparks as well?"
The other champions looked thoughtful, so it appeared Harry was not the only person questioning how to escape the hedges.
"Didn't I mention that in the earlier instructions?" asked Bagman with a frown. He was met with four shaking heads. "Oh dear. I had assumed…no matter. The Triwizard Cup has been enchanted as a Portkey, designed to bring the winner back here in front of the crowd so everyone can know as soon as the winner emerges. I'm glad you saved me from that mistake, Harry. Now, is everyone ready?" The champions nodded, and set their game faces. "Excellent!" Bagman raised his wand to his throat and incanted, "Sonorus."
(Scene break: Despite having slightly more common sense in their heads, the champions perform in the maze as they did in canon.)
From Harry Potter and the Goblet of Fire
"On three, right?" said Harry. "One - two - three -"
He and Cedric both grasped a handle. Instantly, Harry felt a jerk somewhere behind his navel. His feet had left the ground. He could not unclench the hand holding the Triwizard Cup; it was pulling him onward in a howl of wind and swirling color, Cedric at his side.
Harry felt his feet slam into the ground; his injured leg gave way, and he fell forward; his hand let go of the Triwizard Cup at last. He raised his head.
"Where are we?" he said.
Cedric shook his head. He got up, pulled Harry to his feet, and they looked around.
They had left the Hogwarts grounds completely; they had obviously traveled miles -perhaps hundreds of miles - for even the mountains surrounding the castle were gone. They were standing instead in a dark and overgrown graveyard; the black outline of a small church was visible beyond a large yew tree to their right. A hill rose above them to their left. Harry could just make out the outline of a fine old house on the hillside.
A wave of fear ran through Harry, as something about the old house seemed vaguely familiar to him, and he was very sure it wasn't something he had seen at Hogwarts. He looked over at Cedric, who was kneeling by the Triwizard Cup.
"Bagman said we were supposed to return to the front of the crowd, yeah?" asked Cedric bemusedly. "So, is this still part of the task then?"
"I don't think so, Cedric," said Harry, "I have the strangest feeling I've seen this place before, and I know that's not a good thing with my luck. Does the Cup still feel like there's some magic in it?" After a quick wave of his wand over the cup, Cedric nodded.
"All right then, here's what I want you to do," said Harry quickly. "Take a good, long look around this place, then help me over to the Cup. Wherever we are now, Dumbledore will want to know later, and the more perspectives he can get, the better off we all are." At Cedric's slightly quizzical look, Harry managed to say "He's got a Pensieve in his office" before Cedric's face lit up with understanding, and he did as he was told.
As Cedric and Harry were ready to pick the Triwizard Cup back off the ground, Harry spotted a figure walking towards them carrying a bundle. Feeling a sudden pressure building in his forehead, Harry yelled, "Grab the Cup! Now!" As the figure raised a wand, both Harry and Cedric touched a handle and were pulled off by the Portkey.
A tired and pained Harry and Cedric returned to the Hogwarts grounds to find the area in a state of near panic. Only when Ludo Bagman had laid eyes on the pair and announced their arrival did the crowd begin exulting the triumph and the judges and Madam Pomfrey came over to check on the Hogwarts boys.
"Harry, Cedric, what happened?" questioned Professor Dumbledore. "The portkey shouldn't have taken that long to bring you out here. Was there a malfunction?"
Neither boy was coherent enough to respond for a few seconds before Harry's brain reengaged enough to say "Pensieve" and point to his head while looking at Dumbledore. After staring at both Harry and Cedric for a few seconds, Dumbledore and the rest of the judges helped the winners to the Headmaster's office to view the boys' memories of the evening.
After taking everyone through both his and Cedric's memories, and explaining his sense of unease in the graveyard, Harry collapsed into a chair in front of the desk. A very troubled looking Dumbledore thanked Harry and Cedric for their memories and their quick thinking in getting out of the graveyard. Dumbledore conferred with the judges, who decided that the awarding of the Triwizard Cup and the prize money could wait until morning. Bagman and Percy Weasley went to tell the Minister of this development, while Karkaroff and Madame Maxime went to see how their champions were recovering from the maze. Harry and Cedric were escorted by Dumbledore to the Hospital Wing, where Madame Pomfrey was waiting.
"Poppy, could you see to it that these boys each receive a dose of Dreamless Sleep so that they may be rested and ready for tomorrow?" queried Dumbledore politely. She responded in the affirmative, and left to fetch the potion. Harry and Cedric each took a bed, and began to change clothes for bedtime.
"I shall return here one more time tonight, and I ask both of you to wait until I do to take your potions," said Dumbledore softly. "If my hunch is correct, the answer to why the Portkey did not work as I expected will be coming in with me."
Dumbledore spoke briefly with Pomfrey before leaving, so Harry and Cedric sat quietly on their beds, waiting for the Headmaster to return so they could both crash into much-needed sleep. Approximately half an hour after leaving, Dumbledore returned to the Hospital Wing, accompanied by Professor McGonagall, and both professors were levitating unconscious bodies into the wing. One looked like Professor Moody, but without his magical eye or wooden leg. Neither Harry nor Cedric recognized the other, and seeing the condition both boys were in, Dumbledore decided that man's identity could wait until morning, allowing both boys to take the Dreamless Sleep potion and rest peacefully.
The next morning, Harry and Cedric were presented with their winnings during a ceremony in the Great Hall. Harry decided he didn't need the trophy, and Cedric outright refused to take the entire prize, so Harry took the bag of gold back to the Gryffindor table as Cedric brought the Triwizard Cup over to Hufflepuff.
"Very well done, both of you," said Dumbledore proudly. "And it is thanks to your efforts that we were able to determine why the Cup brought you to the incorrect location initially. All year, the person we believed to be Alastor Moody was in fact an impostor taking Polyjuice Potion. The impostor's name is Bartemius Crouch, Junior, and the Ministry believed him to have died at Azkaban Prison many years ago. Therefore, they were very grateful that whatever error allowing him to be released has been corrected, and have awarded both Harry and Cedric Orders of Merlin, Third Class. Another round of applause for them, if you please."
Most of the Great Hall applauded loudly, with the majority of Slytherin House being the only exception.
"It seems another year is coming to a close here at Hogwarts," said Dumbledore. "We would like to thank Durmstrang and Beauxbatons for coming to see us this year, and we hope that you will come visit us again should the notion strike you."
The students from both schools began returning to their respective methods home, and as many goodbyes were exchanged, Beauxbatons and Durmstrang departed Hogwarts as spectacularly as they had arrived.