Prove that for all positive real numbers $a$, $b$ the following inequality holds: \begin{align*} \sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab} \end{align*}When does equality hold?

Let $I$ be the incenter, $A_1$ and $B_1$ midpoints of sides $BC$ and $AC$ of a triangle $\Delta ABC$. Denote by $M$ and $N$ the midpoints of the arcs $AC$ and $BC$ of circumcircle of $\Delta ABC$ which do contain the other vertex of the triangle. If points $M$, $I$ and $N$ are collinear prove that: \begin{align*} \angle AIB_1=\angle BIA_1=90^{\circ} \end{align*}

Find all natural numbers $n$ for which the following $5$ conditions hold: $(1)$ $n$ is not divisible by any perfect square bigger than $1$. $(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$. $(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n$. Then we have that $S(n)+2=d(n)$. $(4)$ $n+3$ is a perfect square. $(5)$ $n$ does not have a prime divisor which has $4$ or more digits.

Initially in every cell of a $5\times 5$ board is the number $0$. In one move you may take any cell of this board and add $1$ to it and all of its adjacent cells (two cells are adjacent if they share an edge). After a finite number of moves, number $n$ is written in all cells. Find all possible values of $n$.