A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be good if it checks simultaneously: \(\bullet a_1<a_2<a_3<\cdots<a_n,\) \(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\) For each \(n\ge 1\), determine how many good lists of \(n\) numbers exist.

Grid the plane forming an infinite board. In each cell of this board, there is a lamp, initially turned off. A permitted operation consists of selecting a square of \(3\times 3\), \(4\times 4\), or \(5\times 5\) cells and changing the state of all lamps in that square (those that are off become on, and those that are on become off). (a) Prove that for any finite set of lamps, it is possible to achieve, through a finite sequence of permitted operations, that those are the only lamps turned on on the board. (b) Prove that if in a sequence of permitted operations only two out of the three square sizes are used, then it is impossible to achieve that at the end the only lamps turned on on the board are those in a \(2\times 2\) square.

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.

Let $ABC$ be an acute triangle and $D, E, F$ are the midpoints of $BC, CA, AB$, respectively. The circle with diameter $AD$ intersects the lines $AB$ and $AC$ at points $P$ and $Q$ , respectively. The lines through $P$ and $Q$ parallel to $BC$ intersect $DE$ at point $R$ and $DF$ at point $S$, respectively. The circumcircle of $DPR$ intersects $AB$ at $X$, the circumcircle of $DQS$ intersects $AC$ in $Y$, and these two circles intersect again point $Z$. Prove that $Z$ is the midpoint of $XY$.

Let $x_1, x_2, \ldots, x_n$ be positive reals; for any positive integer $k$, let $S_k=x_1^k+x_2^k+\ldots+x_n^k$. (a) Given that $S_1<S_2$, show that $S_1, S_2, S_3, \ldots$ is strictly increasing. (b) Prove that there exists a positive integer $n$ and positive reals $x_1, x_2, \ldots, x_n$, such that $S_1>S_2$ and $S_1, S_2, S_3, \ldots$ is not strictly decreasing.