Have you never seen a Western? Typical dialogue:

“Them there critters are mighty jumpy tonight”

And here is a real example from The Legend of Barry Claw

“Them there Injuns sure won’t never forgets…”

Enjoy.

‘Them bones, them bones, them dry bones’ are lyrics from a spiritual, where they originally appear as 'dem', that also appear as 'them', in a song by Alice in Chains.

http://www.metrolyrics.com/them-bones-lyrics-alice-in-chains.html

‘Them’, as opposed to ‘theme’, is the correct word for a group of people or items.

Them! is a 1954 fantasy sci-fi movie.

https://www.google.com.sg/search?q=them&ie=UTF-8&oe=UTF-8&hl=en-sg&client=safari

?

Them?

This is a complete sentence.

At the risk of stating the obvious:

'Them' is the word that starts this sentence.

Them was Van Morrison's band in the 60's.

Them Again was the name of their second album.

Them In Reality was the name of their 1971 album.

At least four sentences on the linked wikipedia page meet your question.

Them Crooked Vultures is a rock supergroup formed in Los Angeles in 2009 by John Paul Jones, Dave Grohl and Josh Homme.

I cannot find the actual competition — if the requirement is "them" but not the word "them", the following would qualify:

Themes are what I enjoy in competitions.

Basically most sentences you'd start off with "It is them" can equally well be started with just "Them".

"This desert has spiky rollypollies. Them you need to fear rather than spiders."

Them and meth are anagrams.

This is a sample sentence.

This form (using a gerund, or noun phrase) hasn't been mentioned yet, and is grammatical albeit awkward:

Them being able to come up with such unusual sentences was a surprise to some but not to others.

(Note that "their" — and perhaps "they" too — is also acceptable as the first word, but by no means obligatory, and in fact less common.)

Here is a sentence beginning with them, from the Cambridge Dictionary.

Them is also used to refer to a person whose sex is not known.

And more:

• Them is the object form of the pronoun they.

• Them is not a suitable word to begin a normal sentence in an SVO language like English.

Several options, all of which sound very antiquated but are probably grammatically acceptable:

Option 1 - Did they specify that the sentence had to be a standalone and understandable paragraph? Because if it were permitted to use it as the beginning of a second sentence, you could easily do something like this, albeit it's antiquated English:

"After Dairyman Wilson had separated the group, he called all of the farmhands between the ages of 18 and 25 to follow him into the fields. Them he gathered by the foot of the old oak tree."

Or:

"John and Iona are expected to perform, but not the others. Them, we do not expect to perform."

Option 2: consider this passage from the King James Bible (Matthew 13:12):

"For whosoever hath, to him shall be given, and he shall have more abundance: but whosoever hath not, from him shall be taken away even that he hath."

Now consider that an analogue of this sentence written to refer to groups of people in the plural, rather than individuals in the singular, and with the meaning changed slightly (for purely illustrative purposes) could be written something like this:

"Them, unto whom much has been given, shall be given more. But those who have not, from them shall be taken away even that they hath."

I'm pretty sure this (or a similar construct) would be acceptable.

Option three: alter the sentence structure of a sentence involving the word "them" and insert a "whom" - e.g., instead of saying:

"We went to the shop with them and they bought a packet of cheerios," You could say: "Them, with whom we went to the shop, bought a packet of cheerios."

I mean, people would think you had just stepped out of the pages of a Thomas Hardy novel, but it would technically be correct. I think.

My grandmother, who lived through the depression as a sixth child in a hardscrabble mining family, had a saying - "Them that has, gets."

'Them heavy people hit me in a soft spot' are lyrics in the Kate Bush song Them Heavy People

And, of course, outside the competition but for the beauty of language: Bob Marley and The Wailers

Them belly full but we hungry
A hungry mob is an angry mob
A rain a fall but the dirt it tough
A pot a cook but the food no 'nough

Them, the barbarians, who are responsible for this heinous act, will be found out and tortured!

This one works for me:

I'm not afraid of spiders, but there were snakes everywhere. Them, I'm afraid of.

Them and you are very difficult to deal with