Soutěžící píšou odpovědi na odpovědní arch.

It can be both. It comes from the verb soutěžit and originally has a participle-like form. That means it can work:

• as an adjective: soutěžící tým = the competing team
• as a noun: soutěžící = contestant

In this sentence, it is being used as a noun.

Both are correct. They are two standard 3rd-person plural present-tense forms of psát = to write.

• píšou = very common, neutral, everyday
• píší = also correct, often felt as more formal or bookish

So Soutěžící píšou... and Soutěžící píší... both work.

It can mean either one. Czech present tense often covers both:

• habitual meaning: Contestants write answers on the answer sheet
• ongoing meaning: Contestants are writing answers on the answer sheet

The context tells you which meaning is intended. English usually has to choose, but Czech often does not.

Odpovědi is accusative plural of odpověď = answer.

It is in the accusative because it is the direct object of píšou: it is the thing being written.

A useful detail: for odpověď, the nominative plural and accusative plural are the same form:

• nominative plural: odpovědi
• accusative plural: odpovědi

So the form itself does not change here, but its function in the sentence is accusative.

Because the preposition na can take different cases depending on the meaning.

Here, na takes the accusative because it means writing something onto a surface or form:

• psát na papír
• psát na formulář
• psát na arch

So na odpovědní arch is the natural wording for onto the answer sheet.

By contrast, na odpovědním archu is locative and would usually describe location, for example:

• Na odpovědním archu jsou dvě chybná jména.
• There are two wrong names on the answer sheet.

It means answer sheet: the sheet or form where someone writes their answers in a test, quiz, or competition.

Also, arch here means sheet. It is not related to the English word arch as in a curved shape. So this is a useful false-friend warning for English speakers.

Because odpovědní is an adjective of the same type as words like:

• jízdní in jízdní řád
• vstupní in vstupní dveře
• platební in platební karta

This adjective pattern has -í in the dictionary form and also in nominative/accusative singular masculine inanimate, which is exactly what we need before arch.

So:

• odpovědní arch = nominative/accusative singular
• odpovědního archu = genitive singular
• odpovědním archu = locative singular

Here arch is accusative singular, but it is a masculine inanimate noun. For masculine inanimate nouns, the accusative singular is often the same as the nominative singular.

So:

• nominative: arch
• accusative: arch

That is why you do not see any extra ending here.

Yes. Czech word order is much more flexible than English word order because grammatical roles are shown by endings as well as context.

The sentence Soutěžící píšou odpovědi na odpovědní arch is a neutral, natural order.

You could also say:

• Odpovědi soutěžící píšou na odpovědní arch.

That version puts more emphasis on odpovědi. The meaning stays basically the same, but the focus changes.

Because Czech does not have articles. There is no direct equivalent of English a/an or the in ordinary Czech grammar.

So soutěžící, odpovědi, and odpovědní arch can be translated in different ways depending on context:

• contestants / the contestants
• answers / the answers
• an answer sheet / the answer sheet

English needs an article; Czech usually does not.