if cuba had entered into a trade agreement with an asian country in 1903 without us approval, which of the following agreements or amendments would that agreement have violated?
the teller amendment
the gentlemen's agreement
the platt amendment
the root-takahira agreement

Correct The Platt Amendment

The Platt Amendment was passed on March 2, 1901, having been introduced in the Senate by Senator Orville Platte of Connecticut.  The Platt Amendment was attached to he 1901 Army Appropriations Bill.  The amendment set forth  seven specific conditions in order for withdrawing US troops that were still in Cuba at the end of the Spanish–American War.  The amendment also stipulated that Cuba would be required to sign a treaty accepting these seven conditions.   The conditions stated by the Platt Amendment were:

1. Cuba could not enter into a treaty with any other foreign country that would "tend to impair" the independence of Cuba.

2. The Cuban government could not "assume or contract any public debt."

3. The the Cuban government agreed "that the United States may exercise the right to intervene for the preservation of Cuban independence" and for maintaining adequate governance there.

4. "That all Acts of the United States in Cuba during its military occupancy thereof are ratified and validated."

5. The Cuban government would carry out plans for public sanitation in the cities of the island to prevent recurrence of epidemics and infectious diseases.

6. The Isle of Pines would be omitted from the proposed boundaries of Cuba.

7. "That to enable the United States to maintain the independence of Cuba, and to protect the people thereof, as well as for its own defense, the government of Cuba will sell or lease to the United States lands necessary for coaling or naval stations at certain specified points to be agreed upon with the President of the United States."

I quoted that last condition in full as a way of showing how the US set itself up in a relationship of dominance over Cuba, asserting "protector" status over the island.

The first provision of the Platt Amendment is the one that would have been violated if (as the question proposed), Cuba sought in 1903 to enter a trade agreement with an Asian country apart from US approval.

norsemen from northern countries had made earlier journeys

the answer is )multiethnic