The inscribed circle of an acute triangle $ABC$ meets the segments $AB$ and $BC$ at $D$ and $E$ respectively. Let $I$ be the incenter of the triangle $ABC$. Prove that the intersection of the line $AI$ and $DE$ is on the circle whose diameter is $AC$(passing through A, C).

For positive integer $n \ge 3$, find the number of ordered pairs $(a_1, a_2, ... , a_n)$ of integers that satisfy the following two conditions For positive integer $i$ such that $1\le i \le n$, $1 \le a_i \le i$ For positive integers $i,j,k$ such that $1\le i < j < k \le n$, if $a_i = a_j$ then $a_j \ge a_k$

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: $a_1=1$ $a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

A sequence of real numbers $a_1, a_2, \ldots $ satisfies the following conditions. $a_1 = 2$, $a_2 = 11$. for all positive integer $n$, $2a_{n+2} =3a_n + \sqrt{5 (a_n^2+a_{n+1}^2)}$ Prove that $a_n$ is a rational number for each of positive integer $n$.

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

Consider $n$ cards with marked numbers $1$ through $n$. No number have repeted, namely, each number has marked exactly at one card. They are distributed on $n$ boxes so that each box contains exactly one card initially. We want to move all the cards into one box all together according to the following instructions The instruction: Choose an integer $k(1\le k\le n)$, and move a card with number $k$ to the other box such that sum of the number of the card in that box is multiple of $k$. Find all positive integer $n$ so that there exists a way to gather all the cards in one box. Thanks to @scnwust for correcting wrong translation.

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$