In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ . The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.

Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that $$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$for all real numbers $x,y$

If $S$ is a set which consists of $12$ elements, what is the maximum number of pairs $(a,b)$ such that $a, b\in S$ and $\frac{b}{a}$ is a prime number?

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$is a perfect square.

In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.

For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality $$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$holds.

Let $r\geq 2$ be a positive integer, and let each positive integer be painted in one of $r$ different colors. For every positive integer $n$ and every pair of colors $a$ and $b$, if the difference between the number of divisors of $n$ that are painted in color $a$ and the number of divisors of $n$ that are painted in color $b$ is at most $1$, find all possible values of $r$.

For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies $$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$Determine the number of divisors of $2024^{2024}$ amongst the sequence.

In a scalene triangle $ABC,$ $I$ is the incenter and $O$ is the circumcenter. The line $IO$ intersects the lines $BC,CA,AB$ at points $D,E,F$ respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC,CA,AB$ at points $X,Y,Z$ respectively. Let the lines $XA_1, YB_1$ and $ZC_1$ intersect $IO$ at points $A_2,B_2,C_2$ respectively. Prove that the circles with diameters $AA_2,BB_2$ and $CC_2$ have a common point.