If $ \frac{a }{b}+ \frac{b}{c}+ \frac{c}{a}$ is integer. show that $ abc$ is perfect cube.

In the acute-angled triangle $ABC$, let $AD$, $D \in BC$ be the $A$-angle bisector. The perpenducular to $BC$ through $D$ and the perpendicular to $AD$ through $A$ meet at $I$. The circle with center $I$ and radius $ID$, intersects $AB$ and $AC$ at $F$ and $E$ respectively. On the arc $FE$, which does not contain $A$, of the circumcircle of $AFE$, consider a point $X$, such that $\frac{XF}{XE}=\frac{AF}{AE}$. Prove that the circumcircles of triangles $AFE$ and $BXC$ are tangent.

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$. Prove that $\max(a_1,a_{2023})\ge 507$.

Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.

Prove that in any triangle the length of the shortest bisector does not exceed three times the radius of the incircle.

Prove that $a=2$ is the greatest real number for which the inequality: $$ \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a $$holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$for all reals $x, y$.

For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$can be written as sum of two squares.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: every term in the sequence is less than or equal to $2^{2023}$, and there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. Ivan Chan Kai Chin, Malaysia