"One of the athletes is not running, but walking."

Translation: Az egyik sportoló nem fut, hanem sétál.

[InkedSkin]

Should athletes not be plural?, i.e.: sportolokat?

[bdori]

No. You have to use the singular form after "egyik". (The word-by-word translation of "one of the athletes" would be "a sportolók közül az egyik" but this unnecessarily long.)

Sportolókat would be plural accusative but here you need the nominative because the athlete is the subject of the sentence.

[AdnanGaleeb]

Does egyik mean none
Or does it mean one of? Or both ?

[ion1122]

By itself, egyik means 'one'. It would mean 'none' only if used together with a negative word such as 'sem'.