Question 1
Question point values reproduce the supplied source labels; no automatic proof grading is implied.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days.
For a positive integer $k$, a set $S$ of positive integers is called $k$-Olympic if both of the following hold:
(i) $S\ne\varnothing$;
(ii) for every $n\in S$, every positive divisor of $(25^n-3^n)k^n$ also belongs to $S$.
Find all positive integers $k$ for which exactly one $k$-Olympic set exists.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days.
For a positive integer $k$, a set $S$ of positive integers is called $k$-Olympic if both of the following hold:
(i) $S\ne\varnothing$;
(ii) for every $n\in S$, every positive divisor of $(25^n-3^n)k^n$ also belongs to $S$.
Find all positive integers $k$ for which exactly one $k$-Olympic set exists.