Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$.

Let $A$ be the number of ways in which the set $\{ 1, 2, . . . , n\}$ can be partitioned into non-empty subsets. Let $B$ be the number of ways in which the set $\{ 1, 2, . . . , n, n + 1 \}$ can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that $A = B$.

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. a) Show that if the initial number written is $0$, then Azambuja cannot reach his goal. b) Find all initial numbers for which Azambuja can achieve his goal.

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called friends is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.