Homer was the epic poet who wrote The Odyssey and The Iliad, but he was not known for his accounts of history, so he would not be the best choice. Plato and Aristotle were philosophers of great renown but neither were concerned with historical accounts, so neither of them would be good choices. Epicurus was concerned with philosophy and the problem of evil, so he too would not be a good choice. Lastly, Herodotus is famous for his account of the Peloponnesian War and he was named the father of history by Cicero, so he would be the best answer.

Archimedes was a philosopher and mathematician; however, he is best known for discovering buoyancy, so he would not be the best choice. Plato and Aristotle were both ancient Greek philosophers, but they are best known for their work regarding the World of Forms and natural philosophy, respectively, so neither of them would be the best choice. Socrates pioneered the dialectic method and had some thoughts about living the good life, but he was not known for an interest in the problem of evil, so he would not be the best answer here. Lastly, Epicurus was concerned both with the good life and the problem of evil, so he would be the best choice here.

Archimedes was a mathematician and philosopher, however he was most famous for his discovery of buoyancy, so he would not be the best answer. Plato and Aristotle were philosophers but did not write extensively regarding mathematics, so they would not be good answers here. Socrates was also a philosopher and not a mathematician, so he would not be a good answer. Euclid wrote Elements -the premiere geometry textbook for millennia- and normal geometry is named after him, so he would be the best choice.

While Athenian culture exuded some of the other features, competition was a hallmark of Athens. This is evident from depictions of the Olympic games and many other frequent small sporting events found on Greek pottery and architecture. 

The Ancient Greek word polis refers to the large city-states of Greece.

Athens formed the Delian League with surrounding Greek city-states in order to better coordinate defense against external forces (such as the Persians). The other answers are incorrect because they contain terms not relevant to this part of Ancient Greek history.

The two "superpowers" of Greece, Athens and Sparta, warred against each other in the Peloponnesian Wars during the mid-fifth century BCE. The war resulted in a temporary supremacy of the Spartans in Greece. The other answers are incorrect because they refer to other nearby factions not involved in the conflict.

Sparta was a militaristic and relatively secluded city-state, with a large social emphasis on loyalty and duty. Naval power was not emphasized as much in Sparta as compared to other Greek city-states - thus this answer is correct. The other answers do pertain to accurate aspects of Sparta.

The Punic Wars were fought between Rome and Carthage. Carthage, with its extremely powerful navy, were expanding their empire to the East, as Rome secured the Italian peninsula and began greater expansion via land. In the second Punic War, Hannibal lead the Carthagian army over the alps to confront the Roman forces.

The Julio-Claudians was the first Roman dynasty, started in 27 BCE with Augustus and ended in 68 CE with the assassination of Nero. The Julio-Claudians set the tone for the first few eras of Roman rule; good emperors tried to behave like Augustus, bad ones tended to err on the side of Caligula or Nero. In this way, the Julio-Claudians function as a lesson in the power, capability and debauchery of the emperors of Rome.