The correct answer is option 1, River Plate. Buenos Aires and Montevideo are two major cities in South America, located on opposite sides of the River Plate. The River Plate, also known as the Rio de la Plata, is a large estuary formed by the confluence of the Uruguay and Paraná rivers.

Option 2, Orinoco River, is incorrect because the Orinoco River is located in northern South America, primarily flowing through Venezuela.

Option 3, Purus River, is incorrect because the Purus River is located in the Amazon Rainforest region of Brazil.

Option 4, Madeira River, is incorrect because the Madeira River is a major tributary of the Amazon River, flowing through Brazil and Bolivia.

Therefore, the correct answer is option 1, River Plate, as Buenos Aires and Montevideo are situated across the banks of this estuary.