Let $ABCD$ be a convex quadrilateral. Construct $S$ and $T$ on the side $AD$ and $AB$ respectively such that $AS=AT$. Construct $U$ and $V$ on the side $BC$ and $CD$ respectively such that $CU=CV$. Assume that $BT=BU$ and $ST, UV, BD$ are concurrent, prove that $AB+CD=BC+AD$.

Find all pairs of positive integers $(m,n)$ such that $\frac{m^5+n}{m^2+n^2}$ and $\frac{m+n^5}{m^2+n^2}$ are integers.

Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$for all positive reals $x$ and $y$.

In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, $$x(c)y(c)\geq184.$$Where $x(c)$ is the number of purple cells that lie in the same row as $c$, and $y(c)$ is the number of purple cells that lie in the same column as $c$. Find the least possible number of cells that are colored purple.

Let $ABC$ be a scalene triangle. Let $H$ be its orthocenter and $D$ is a foot of altitude from $A$ to $BC$. Also, let $S$ and $T$ be points on the circumcircle of triangle $ABC$ such that $\angle BSH=\angle CTH=90^{\circ}$. Given that $AH=2HD$, prove that $D,S,T$ are collinear.

Find all positive integers $n$ such that the elements of $$\{1,2,...,2n+1\}-\{n+1\}$$can be partitioned into two groups with the same number of elements and the same sum of their elements.

Let $m$ and $n$ be positive integers for which $n\leq m\leq 2n$. Find the number of all complex solutions $(z_1,z_2,...,z_m)$ that satisfy $$z_1^7+z_2^7+...+z_m^7=n$$Such that $z_k^3-2z_k^2+2z_k-1=0$ for all $k=1,2,...,m$.

Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively. Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.

Prove that for all positive integers $n$, there exists a sequence of positive integers $a_1,a_2,\dots,a_n$ and $d_1,d_2,\dots,d_n$ satisfying all of the following three conditions. $\binom{2a_i}{a_i}$ is divisible by $d_i$ for all $i=1,2,\dots,n$ $d_{i+1}=d_i+1$ for all $i=1,2,\dots, n-1$ $d_i\neq m^k$ for all $i=1,2,\dots, n$ and positive integers $m$ and $k$ such that $k\geq 2$

Find the maximum value of \[abcd(a+b)(b+c)(c+d)(d+a)\]such that $a,b,c$ and $d$ are positive real numbers satisfying $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+\sqrt[3]{d}=4$