Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape? ( Proposed by Vlad Spǎtaru, Romania)

Let, $\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}$. Prove that for $n_i \in \mathbb{R}^+$ $$\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}$$ Proposed by Kang Taeyoung, South Korea

Let $ABC$ be an acute triangle and $H$ be its orthocenter. Let $E$ be the foot of the altitude from $C$ to $AB$, $F$ be the foot of the altitude from $B$ to $AC$. Let $G \neq H$ be the intersection of the circles $(AEF)$ and $(BHC)$. Prove that $AG$ bisects $BC$. Proposed by Kang Taeyoung, South Korea

Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer. Proposed by Prajit Adhikari, Nepal