Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.

Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$Prove that $YI$ is the angle bisector of $XYA$.

Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$. Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$, such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$

Given a positive integer $n\geq 2$ and positive real numbers $a_1, a_2, \ldots, a_n$ with the sum equal to $1$. Let $b = a_1 + 2a_2 + \ldots + n a_n$. Prove that $$\sum_{1\leq i < j \leq n} (i-j)^2 a_i a_j \leq (n-b)(b-1).$$

Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets $$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$have at most $3$ common elements.

For any real numbers $a$ and $b>0$, define an extension of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$. We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$. Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$, there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$, their extensions covers $A$.