What happens to the Triwizard Cup when the tournaments reoccur? Is it passed around to the next following champions? Or is a new one made?
It seems to be one cup that’s passed along.
Though it’s not stated explicitly, there seems to be only one Triwizard Cup. Dumbledore mentions that he knows students will be eager to bring the Triwizard Cup to Hogwarts, which implies that it’s a single cup. It’s referred to as ‘the’ Triwizard Cup, which implies that only one exists, and Dumbledore’s statement implies that Hogwarts doesn’t already have one or more Triwizard Cups.
“Eager though I know all of you will be to bring the Triwizard Cup to Hogwarts,’ he said, ‘the Heads of the participating schools, along with the Ministry of Magic, have agreed to impose an age restriction on contenders this year.”
- Harry Potter and the Goblet of Fire, Chapter 12 (The Triwizard Tournament)
If there was indeed a new cup made for every tournament, it seems highly unlikely that Hogwarts wouldn’t have earned at least one Triwizard Cup over the many previous years the Triwizard Tournament was held before it was discontinued, and thus already have at least one Triwizard Cup at Hogwarts before then. The Triwizard Tournament began around seven hundred years ago, and was held every five years before it was discontinued due to its high death toll.
“The Triwizard Tournament was first established some seven hundred years ago, as a friendly competition between the three largest European schools of wizardry – Hogwarts, Beauxbatons and Durmstrang. A champion was selected to represent each school, and the three champions competed in three magical tasks. The schools took it in turns to host the Tournament once every five years, and it was generally agreed to be a most excellent way of establishing ties between young witches and wizards of different nationalities – until, that is, the death toll mounted so high that the Tournament was discontinued.”
- Harry Potter and the Goblet of Fire, Chapter 12 (The Triwizard)
The Triwizard Tournament was still being held in 1792, so that would mean that the three schools had centuries of holding the Triwizard Tournament every five years.
“Well, the Heads of the participating schools are always on the panel,’ said Hermione, and everyone looked around at her, rather surprised, ‘because all three of them were injured during the Tournament of 1792, when a cockatrice the champions were supposed to be catching went on the rampage.”
- Harry Potter and the Goblet of Fire, Chapter 15 (Beauxbatons and Durmstrang)
Presumably during that time, Hogwarts would’ve won the Triwizard Tournament at least once, so should have at least one Triwizard Cup if a new one was made for each tournament and given to the winning school. Therefore, it’s most logical to conclude from the information given that there’s only one Triwizard Cup, which the winning school holds until the next Triwizard Tournament is held. There’s precedent for this in the wizarding world, since the Hogwarts Quidditch Cup is passed along each year to the winning team’s Head of House - McGonagall mentioned that she’s used to having the Quidditch Cup in her study and she doesn’t want to hand it over to Snape.
“Nobody could quite believe their ears until she looked directly at Harry and Ron and said grimly, ‘I’ve become accustomed to seeing the Quidditch Cup in my study, boys, and I really don’t want to have to hand it over to Professor Snape, so use the extra time to practise, won’t you?”
- Harry Potter and the Order of the Phoenix, Chapter 19 (The Lion and the Serpent)
It seems very likely from what’s known about the Triwizard Cup that it’s a singular cup passed along in the same way as the Quidditch Cup at Hogwarts is. Additionally, even in the Muggle world, “Cups” traditionally tend to be one item passed along to each new winner. The Stanley Cup, the World Cup, the Ryder Cup, the Grey Cup, and America’s Cup are all singular cups passed along to each new winner. This meaning of ‘cup’ is especially apparent when contrasted with ‘trophies’, which tend to be made anew for each event and kept permanently by the winners, like the Vince Lombardi Trophy, the Commissioner’s Trophy, and the Larry O’Brien Championship Trophy. Wizarding sports seem to operate along these same rules, since the Quidditch Cup at Hogwarts is passed on to the new winner each year.
We always say during World cup(Cricket or any sports), Bring the world cup to our country, that doesn't mean there is only one cup, but that means we should win the championship this year.
There is no clear evidence for what happens to cup, but we can't make conclude like it's the same cup, I guess.
The Triwizard Cup is a unique object made hundreds of years before the events of the Harry Potter books. After each Triwizard Tournament, the cup was held as a mark of honor at whichever school was victorious until it was returned for the next tournament.