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.
Let $n$ be a positive integer. A country has $8n+3$ airports. Between any two airports there is either a direct flight route or no direct route. Whenever two airports have no direct route between them, their numbers of direct routes differ by exactly $2$. Determine the minimum total number of direct routes.
Let $ABC$ be an acute scalene triangle with altitudes $AD,BE,CF$. From $A$, drop perpendiculars to the lines $EF,FD,DE$, with feet $X,Y,Z$, respectively. Let the line $BZ$ meet the circle $(BDY)$ again at $P$, and let $CY$ meet $(CDZ)$ again at $Q$. Prove that $X$ has equal powers with respect to the circles $(YFP)$ and $(ZEQ)$.
Let $ABC$ be a triangle, and let $O$ be the midpoint of $BC$. Draw the tangents $AE,AF$ to the circle $(O)$ with diameter $BC$, where $E,F\in(O)$. The rays $AE,AF$ meet $BC$ at $K,L$, respectively. Let $KF,LE$ meet $(O)$ again at $M,N$, respectively. The circumcircle of triangle $MON$ meets the circles with diameters $AB,AC$ again at $X,Y$, respectively. Prove that $\angle XAB=\angle YAC$.
Let $k,n$ be positive integers with $k\lt n$. Find all real-coefficient polynomials $P$ of degree $kn$ with leading coefficient $1$ such that
$$Q(x)=P(x^n+1)-(P(x))^n$$
has degree at most $kn(n-1)$.
Let $\mathcal H$ be a family of nonempty subsets of $\{1,2,3,\ldots,2027\}$ with the following property: for every $A\in\mathcal H$ and every subset $B\subset A$, one has $B\in\mathcal H$. Let $\ell_{\mathcal H},c_{\mathcal H}$ denote, respectively, the numbers of sets in $\mathcal H$ having an even number of elements and an odd number of elements. Prove that
$$\ell_{\mathcal H}-c_{\mathcal H}\le\binom{2026}{1013}.$$