Starfleet Academy entrance exam
The Starfleet Academy entrance exam was an exam administered to candidates for Starfleet Academy. 

Federation citizens were eligible for the exam. For non-Federation citizens, eligibility was considered with a letter of reference from a command level Starfleet officer. For example, a Ferengi, Nog, became a Starfleet cadet with the backing of Commander Benjamin Sisko. ()

The first requirement for taking the exam was to pass a preliminary test. Finalists were selected from groups of high-scoring candidates to take the entrance exam. A certain number of those with the highest scores were then admitted into the Academy. Candidates who did not succeed were allowed to re-take the exam after a period of time, typically an Earth year.

Exam facilities were located throughout the Federation. One of them was in a Federation starbase on Relva VII. ()

The exam itself was designed to look for certain qualities a person displayed, including: integrity, intelligence, courage, imagination, and leadership qualities. Candidates were told to expect the unexpected during this exam.

Examples of tests which comprised the entrance exam included:
Dynamic relationships test – a spinning 3D cube consisting of a grid of blue dots in a 6×6×6 configuration, totaling 216 individual points. Surrounding the cube were 216 red dots flying around in a seemingly random pattern. To complete this test each of the red dots must match up with a blue dot. The solution required discovering the appropriate rotation factor and vector coordinates.
Hyperspace physics test – among others, it featured a trick question: "If the matter and antimatter tanks on a  starship are nine-tenths depleted, calculate the intermix ratio necessary to reach a starbase one hundred light years away at warp factor 8." It was a trick question because there is only one matter/antimatter intermix ratio, that being one to one.
Psych test – this test allowed the candidate face his or her deepest fears. The test itself was tuned to the experiences of the candidate through an analysis of the test taker's psychological profile. Wesley Crusher found this section of the test the most difficult and fearsome.
Other cultures and species test – not always announced in advance, its purpose was to test the candidate's reaction to a member of another species. The reaction of the candidate was important.

The flux coordinating sensor was part of the apparatus on which the candidates performed some of the tests. ()

Starfleet Academy