A point $P$ lies inside $\usepackage{gensymb} \angle ABC(<90 \degree)$. Show that there exists a point $Q$ inside $\angle ABC$ satisfying the following condition: For any two points $X$ and $Y$ on the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ respectively satisfying $\angle XPY = \angle ABC$, it holds that $\usepackage{gensymb} \angle XQY = 180 \degree - 2 \angle ABC.$

Let $d(n)$ be the number of divisors of $n$. Show that there exists positive integers $m$ and $n$ such that there are exactly 2024 triples of integers $(i, j, k)$ satisfying the following condition: $0<i<j<k \le m$ and $d(n+i)d(n+j)d(n+k)$ is a multiple of $ijk$

Consider any sequence of real numbers $a_0$, $a_1$, $\cdots$. If, for all pairs of nonnegative integers $(m, s)$, there exists some integer $n \in [m+1, m+2024(s+1)]$ satisfying $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$, say that this sequence has repeating sums. Is a sequence with repeating sums always eventually periodic?

Show that there are infinitely many positive odd integers $n$ such that $n^5+2n+1$ is expressible as a sum of squares of two coprime integers.

For each positive integer $n>1$, if $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$($p_i$ are pairwise different prime numbers and $\alpha_i$ are positive integers), define $f(n)$ as $\alpha_1+\alpha_2+\cdots+\alpha_k$. For $n=1$, let $f(1)=0$. Find all pairs of integer polynomials $P(x)$ and $Q(x)$ such that for any positive integer $m$, $f(P(m))=Q(f(m))$ holds.

For a given positive integer $n$, there are a total of $5n$ balls labeled with numbers $1$, $2$, $3$, $\cdots$, $n$, with 5 balls for each number. The balls are put into $n$ boxes, with $5$ balls in each box. Show that you can color two balls red and one ball blue in each box so that the sum of the numbers on the red balls is twice the sum of the numbers on the blue balls.

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions: $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$ For all real number $x$, $f(g(x))=g(f(x))=0$ For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$ For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$

Let $\omega$ be the incircle of triangle $ABC$. For any positive real number $\lambda$, let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$. Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\angle C$ closer to $BC$. Similarly define $Y$ and $Z$. Let $\epsilon = \frac{1}{2024}$. Show that the circumcircle of triangle $XYZ$ lies inside $\omega_{1-\epsilon}$. Note. Weaker results with smaller $\epsilon$ may be awarded points depending on the value of the constant $\epsilon <\frac{1}{2024}$.