The answer to your question is complicated (or at least nuanced)...
It is illegal according to US federal law to ship alcoholic beverages via the United States Postal Service. According to the terms of service of UPS and Fedex, alcoholic beverages may only be shipped from a licensed (to manufacture and/or distribute) entity. Some states allow unlicensed individuals to receive alcoholic beverages, others don't.
So basically, even if you lived in the US, you couldn't ship homebrew to a competition.
But wait, you say... lots of people ship homebrew to competitions. How does that work? The typical dodge is to mark the shipment "Yeast Samples" and ship away. This dodge may work with UPS and Fedex, but do not try it with the postal service. Shipping across international boundaries, this subterfuge is probably not a good idea (at least I wouldn't do it).
Most of this weirdness comes from prohibition-era laws and regulations (prohibition ended in December of 1933). In fact, the entire alcoholic beverage distribution system in most of the US is screwed up due to these kinds of laws and regulations.
Bottom line is you probably won't be able to legally ship your homebrew to a competition in the US.
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