Final answer:
The countries that raced to build up their navies after 1900 were primarily motivated by militarism, competition, nationalism, and imperialism.
Step-by-step explanation:
The countries that raced to build up their navies after 1900 were primarily motivated by **militarism**, which is the belief or desire of a government or people that a country should maintain a strong military capability and be prepared to use it aggressively to defend or promote national interests. This militaristic mindset was closely tied to **competition** between nations, as they sought to showcase their strength and power. Additionally, the desire for **nationalism** and the pursuit of imperialist ambitions also played a role in this race for naval expansion.