Step 1: Read the question.
After the Ottoman Empire fell, we must find the two countries that wanted to control the Black Sea and Mediterranean region.
Step 2: Why this region mattered.
The Ottoman lands linked Europe, Asia, and Africa and controlled key sea routes. After the empire weakened, European powers raced to grab influence there.
Step 3: Look at the choices.
Germany and Austria were focused on central Europe. France and Poland were not the main rivals over these seas. Russia and Turkey were part of the older Ottoman story but Turkey itself was the fallen power.
Step 4: The real contenders.
Italy wanted to expand its power in the Mediterranean, and Britain wanted to protect its trade route to India through these waters.
Step 5: Match the pair.
So the two countries seeking control here were Italy and Britain.
Step 6: Conclude.
The answer is Italy and Britain.
\[ \boxed{\text{Italy and Britain}} \]