For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. Define $a_n=\dfrac1{4^{[-\log_4 n]}}$ for $n\ge1$, and put
$$b_n=\frac1{n^2}\left(\sum_{k=1}^na_k-\frac1{a_1+a_2}\right).$$
(a) Find a real-coefficient polynomial $P(x)$ such that $b_n=P(a_n/n)$ for every $n\ge1$.
(b) Prove that there is a strictly increasing sequence of positive integers $(n_k)$ such that $\displaystyle\lim_{k\to\infty}b_{n_k}=2024/2025$.
Find all real-coefficient polynomials $P(x),Q(x)$ such that for every real number $a$, the number $P(a)$ is a root of
$$x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.$$
Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$ be the center of the circle passing through $C$ and tangent to $AB$ at $A$; define $B'$ as the center of the circle through $A$ tangent to $BC$ at $B$, and $C'$ as the center of the circle through $B$ tangent to $CA$ at $C$.
(a) Prove that the area of $A'B'C'$ is at least the area of $ABC$.
(b) Let $X,Y,Z$ be the perpendicular projections of $O$ onto $A'B',B'C',C'A'$, respectively. Suppose the circumcircle of $XYZ$ meets these three lines again at $X',Y',Z'$, respectively, with $X'\ne X,Y'\ne Y,Z'\ne Z$. Prove that $AX',BY',CZ'$ are concurrent.
Place $k$ marbles in the cells of a $2024\times2024$ square board so that each cell contains at most one marble and no two marbles occupy adjacent cells. Two cells are adjacent if they share a side.
(a) For $k=2024$, give such a placement in which moving any marble to any adjacent cell produces a placement no longer satisfying both conditions.
(b) Find the largest $k$ such that, for every placement satisfying the two conditions, some marble can be moved to an adjacent cell so that no two marbles are adjacent in the new placement.
For a polynomial $P(x)$, define $P_1(x)=P(x)$ and $P_{j+1}(x)=P(P_j(x))$ for $1\le j\le2023$, for all real $x$. Let $a\gt 2$ be real. Does there exist a real-coefficient polynomial $P(x)$ such that for every $t\in(-a,a)$, the equation $P_{2024}(x)=t$ has exactly $2^{2024}$ distinct real roots?
For a positive integer $n$, let $\tau(n)$ be the number of its positive divisors.
(a) Solve $\tau(n)+2023=n$ in positive integers $n$.
(b) Prove that there are infinitely many positive integers $k$ for which exactly two positive integers $n$ satisfy $\tau(kn)+2023=n$.
In space, let $D$ be a convex polyhedron such that an even number of edges meets at each vertex. Choose one face $F$ of $D$. Assign a positive integer to each edge, with the property that for every face other than $F$, the sum of the integers on its boundary edges is divisible by $2024$. Prove that the sum on the boundary edges of $F$ is also a positive integer divisible by $2024$.