Chapter 20: The First Task

Simplified Chinese (Mandarin: China)
第一个项目 Dì-yī-ge xiàngmù	第一 dì-yī = 'no. 1, first'. 个 ge = counter/classifier 项目 xiàngmù = 'item, project, event'.	The First Item
Traditional Chinese (Mandarin: Taiwan)
第一項任務 Dì-yī-xiàng rènwu	第一 dì-yī = 'no. 1, first'. 項 xiàng = counter/classifier 任務 rènwu = 'assignment, mission, task'.	The First Assignment
Japanese
第一の課題 Dai-ichi no kadai	第一 dai-ichi = 'no. 1, first'. の no = connecting particle 課題 kadai = 'subject, theme, task, exercise, problem'.	The first task
Korean
첫 번째 시험 Cheos beonjjae siheom		The First Examination
Vietnamese (Chinese characters show etymology)
Bài thi đầu tiên	bài thi = 'examination paper'. đầu tiên (頭先) = 'first'.	The First Examination
Mongolian (new)
Анхны даалгавар Ankhnii daalgavar	анх ankh = 'first' (-(н)ий -(n)ii Genitive form). даалгавар daalgavar = 'task, job, mission'.	The First Task

Task

This is the first task in the Triwizard Tournament.

The Mainland Chinese translation uses the general word 项目 xiàngmù meaning 'item', 'project', or 'event'. (The classifier/counter used is 个 ge, the most common, all-purpose classifier found in Chinese.)
The Chinese translator from Taiwan uses the word 任務 rènwu, 'mission, task' which is used not only of assignments given to secret agents but also quite ordinary tasks given to staff to perform. (The classifier/counter used here is the rather more specialised 項 xiàng, meaning 'item'. This is a classifier used specifically for things that are itemised.)
The Japanese 課題 kadai is a fairly broad term used for tasks, exercises, subjects, problems, and themes.
The Vietnamese treats the tournament as a kind of examination or test (bài thi, examination paper).
The Mongolian uses the word даалгавар daalgavar, meaning 'task, job, mission', and also used for something as mundane as homework.

First

Chinese and Japanese use 第一 dì-yī (Chinese) dai-ichi (Japanese) for 'first' or 'no. 1'.
Vietnamese đầu tiên is formed of Chinese roots (頭先 = 'head first'), but this is not found in Mandarin.
The Mongolian term for 'first' is анхны ankhnii, from анх ankh 'first'.

(Detailed notes on the chapter can be found at Harry Potter Lexicon)