Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$

Let $ABCD$ be an inscribed quadrilateral. Let the incenters of $BAD$ and $CAD$ be $I$ and $J$ respectively. Let the intersection point of the line that passes through $I$ and perpendicular to $BD$ and the line that passes through $J$ and perpendicular to $AC$ be $K$. Prove that $KI=KJ$

Let $m,n$ be relatively prime positive integers. Prove that the numbers $$\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}$$cannot be integer at the same time.

Let $x_1,x_2,\dots,x_{31}$ be real numbers. Then find the maximum value can $$\sum_{i,j=1,2,\dots,31, \; i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1,2,\dots,31}{\lfloor x_i^2 \rfloor } \right)$$achieve. P.S.: For a real number $x$ we denote the smallest integer that does not subseed $x$ by $\lceil x \rceil$ and the biggest integer that does not exceed $x$ by $\lfloor x \rfloor$. For example $\lceil 2.7 \rceil=3$, $\lfloor 2.7 \rfloor=2$ and $\lfloor 4 \rfloor=\lceil 4 \rceil=4$