Vietnam TST – T's Lab

Day 1 (March 26, 2025).

Time allowed: 270 minutes.

Problem 1. For a positive integer $k$ , a set $S$ of positive integers is called a $k$ -Olympic set if it satisfies the following conditions simultaneously:
(i) $S \neq \emptyset$ .
(ii) For every $n \in S$ , all positive divisors of $(25^n - 3^n)k^n$ also belong to $S$ .
Find all positive integers $k$ such that there is exactly one such $k$ -Olympic set.

Problem 2. Let $n$ be a positive integer, and in a country, there are $8n+3$ airports. Between any two airports, there is either a direct flight or not. Given that if there is no direct flight between two airports, the difference in the number of direct flights from these two airports is exactly 2. Determine the minimum possible total number of direct flights.

Problem 3. Let $ABC$ be an acute non-isosceles triangle with altitudes $AD, BE, CF$ . From vertex $A$ , drop perpendiculars to the lines $EF, FD, DE$ , denoted as $X, Y, Z$ respectively. Let the line $BZ$ intersect the circumcircle of triangle $BDY$ again at $P$ , and let the line $CY$ intersect the circumcircle of triangle $CDZ$ again at $Q$ . Prove that point $X$ has the same power with respect to the two circles $(YFP)$ and $(ZEQ)$ .