Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] Proposed by Otgonbayar Uuye.

We call a triangle consisting of three vertices of a pentagon big if it's area is larger than half of the pentagon's area. Find the maximum number of big triangles that can be in a convex pentagon. Proposed by Gonchigdorj Sandag

Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\] Proposed by Otgonbayar Uuye.

Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$. $(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$. $(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$. Proposed by Otgonbayar Uuye.

Let $ABC$ be an acute-angled triangle and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to the sides $AC$ and $AB$ respectively. Suppose $AD$ is the diameter of the circle $ABC$. Let $M$ be the midpoint of $BC$. Let $K$ be the imsimilicenter of the incircles of the triangles $BMF$ and $CME$. Prove that the points $K, M, D$ are collinear. Proposed by Bilegdembrel Bat-Amgalan.

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.