Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions: $(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater. $(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal. Find all possible values of $n$.

Let $\triangle ABC$ be an acute triangle, where $H$ is the orthocenter and $D,E,F$ are the feet of the altitudes from $A,B,C$ respectively. A circle tangent to $(DEF)$ at $D$ intersects the line $EF$ at $P$ and $Q$. Let $R$ and $S$ be the second intersection points of the circumcircle of triangle $\triangle BHC$ with $PH$ and $QH$, respectively. Let $T$ be the point on the line $BC$ such that $AT\perp EF$. Prove that the points $R,S,D,T$ are concyclic.

For all $n\ge2$ positive integer, let $f(n)$ denote the product of all distinct prime divisors of $n$. For example, $f(5)=5$, $f(8)=2$, and $f(12)=6$. Given a sequence ${a_n}$, where $a_1\ge2$, defined as follows: $$a_{n+1}=a_n+f(a_n)$$ Show that for any prime $p$, there exists a term $a_k$ in the sequence such that $p|a_k$.

Let $n$ be a positive integer, and let $1=d_1<d_2<\dots < d_k=n$ denote all positive divisors of $n$, If the following conditions are satisfied: $$ 2d_2+d_4+d_5=d_7$$$$ d_3 d_6 d_7=n$$$$ (d_6+d_7)^2=n+1$$ find all possible values of $n$.

Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$: $$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .

Let $m,n\ge2$ be positive integers. On an $m\times n$ chessboard, some unit squares are occupied by rooks such that each rook attacked by odd number of other rooks. Determine the maximum number of rooks that can be placed on the chessboard.