One common SEO myth is that Google has some type of "duplicate content penalty". That's simply not true. After you write a good original article and publish it to your site, its hard to stop people from copying it and re-publishing it. If there really was a penalty for duplicate content, it would open the door for black hat SEOs to simply copy an article from their competitor and blast it out on different article syndication networks to get their competition penalized. That's just one of the problems with search engines creating that type of penalty, and Google knows this.
However, one thing that Google DOES have is a duplicate content filter. What this means is if they find 100 versions of the same article on different sites they don't want results 1 to 100 the return the same article. This would be a bad experience for Google visitors, who are the people they want to keep happy. Instead, they only display one version of the article, and they pick it from the site with the highest trust and authority (pagerank, backlinks, etc.).
From what you said in your example, my guess is your competitor probably has a site with lots of backlinks and authority. That would allow them to rank for articles even if they're duplicates. My advice to you is to keep writing good quality original content. Over the long term that is the best SEO strategy. In addition, you will want to increase your backlinking efforts to generate more links to your site and increase your site authority. This is where most SEO efforts fizzle out because it really is a lot of work building quality backlinks. The results are definitely worth it though!