What prompted the u. s. formally to state its open door policy 1899 and 1900?

for protection of equal privileges among countries trading with China and in support of Chinese territorial and administrative integrity.

hope thus helps <3

The United States had taken possession of the Philippines (as well as Guam and Puerto Rico) in 1898, after winning the Spanish-American War.   Thus US interest in Asia was heightened.

At the same time, other nations had begun competing for "spheres of influence" in trade access with China.

Further detail:

The Open Door policy was issued by the United States in 1899-1900 as a series of dispatches from the US Secretary of State to other nations that had trading interests in China --  Great Britain, Germany, France, Italy, Japan, and Russia.  The policy reasserted earlier agreements that all countries should have equal access to ports in China, without undue preference to "spheres of influence" for one nation or another.  The United States was seeking to maintain an equal footing with other nations in the access to trade in China.

the main objectives of the progressive movement were eliminating problems caused by industrialization, urbanization, immigration, and political corruption.