Does operating an online business that specializes in selling second-hand products for a single brand (using their model names to simplify the search of available used products) could cause legal problems?
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A trade mark exists to identify the goods and services provided by the business that owns the trade mark and infringement occurs when you use it is such a way to cause confusion that your goods and services are their goods and services.
A website to sell second-hand Toyota's (say) does not cause such confusion - the brand identifies their goods and cannot be confused with your service (the second hand marketplace).