Well mostly, they don't actually arrive until the 30th October:
The delegations from Beauxbatons and Durmstrang will be arriving at 6 o'clock on Friday 30th of October.
Harry Potter and the Goblet of Fire - p.207 - Bloomsbury - Chapter 15, Beauxbatons and Durmstrang
But they stayed till the end.
This is during the end of term feast:
Dumbledore looked from Madame Maxime and Hagrid, to Fleur Delacour and her fellow Beauxbatons students, to Viktor Krum and the Durmstrangs at the Slytherin table. Krum, Harry saw, looked wary, almost frightened, as though he expected Dumbledore to say something harsh.
Harry Potter and the Goblet of Fire - p.627 - Bloomsbury - Chapter 37, The Beginning
As for where they had their lessons, there's clearly some disagreement, and it isn't completely clear, because we aren't in their year, so Rowling doesn't bother showing us their schooling arrangements - why would she? But I can't see Maxime or Karkaroff teaching an entire year's syllabus, across all classes, in their little carriage/boat - they seemed to just be the sleeping arrangements. Also what's the point of keeping the students there, if they're not going to attend Hogwarts classes? They might just as well go home.
But do I have anything other than an appeal to Occam's razor?
Well, yes, I think the clearest example is:
Harry glimpsed Fleur Delacour from time to time in the corridors; she looked exactly as she always did, haughty and unruffled.
Harry Potter and the Goblet of Fire - pp.281-2 - Bloomsbury - Chapter 19, The Hungarian Horntail
It's very difficult to see what she'd be doing in the corridors if she was just popping up to Hogwarts for meals. But here's a few other pieces of evidence.
'He's really nice, you know,' she said. 'He's not at all like you'd think, coming from Durmstrang. He likes it much better here, he told me.'
Harry Potter and the Goblet of Fire - p.386 - Bloomsbury - Chapter 24, Rita Skeeter's Scoop
This seems like a pretty meaningless statement if he's being taught by Karkaroff on his boat, doesn't it?
The Hogwarts staff, demonstrating a continued desire to impress the visitors from Beauxbatons and Durmstrang, seemed determined to show the castle at its best this Christmas. When the decorations went up, Harry noticed that they were the most stunning he had yet seen inside the school. Everlasting icicles had been attached to the banisters of the marble staircase; the usual twelve Christmas trees in the Great Hall were bedecked wth everything from luminous holly berries to real, hooting, golden owls, and the suits of armour had all been bewitched to sing carols whenever anyone passed them. It was quite something to hear 'Oh Come, All Ye Faithful' sung by an empty helmet that only knew half the words. Several times, Filch the caretaker had to extract Peeves from inside the armour, where he had taken to hiding, filling in the gaps in the songs with lyrics of his own invention, all of which were very rude.
Harry Potter and the Goblet of Fire - p.344 - Bloomsbury - Chapter 22, The Unexpected Task
Certainly makes it seem to me like the guests do more than just come up to Hogwarts for meals. And what would be the point? If you can teach Potions and Charms in there, you can surely get a meal delivered there. Also, don't forget, the age limit is seventeen, which means sixth and seventh years can participate, so there's not necessarily just one year group.
Anyway, I haven't yet found a smoking gun, but both Fleur and Krum seem to know the castle fairly well.
'Vell, ve have a castle also, not as big as this, nor as comfortable, I am thinking,' he was telling Hermione. 'Ve have just four floors, and the fires are lit only for magical purposes. But ve have grounds larger even than these - though in vinter, ve have very little daylight, so ve are not enjoying them. But in summer ve are flying every day, over the lakes and the mountains -'
Harry Potter and the Goblet of Fire - p.363 - Bloomsbury - Chapter 23, The Yule Ball
'Zis is nothing,' she said dismissively, looking around at the sparkling walls of the Great Hall. 'At ze Palace of Beauxbatons, we 'ave ice sculputres all around ze Dining Chamber at Chreestmas. Zey do not melt, of course ... zey are like 'uge statues of diamond, glittering around ze place. And ze food is seemply superb. And we 'ave choirs of wood-nymphs 'oo serenade us as we eat. We 'ave none of zis ugly armour in ze 'alls, and eef a poltergeist ever entaired into Beauxbatons, 'e would be expelled like zat.'
Harry Potter and the Goblet of Fire - p.364 - Bloomsbury - Chapter 23, The Yule Ball