UZ-MO 2016

$\omega$ is circumcircle of triangle $ABC$ and $BB_1, CC_1$ are bisectors of ABC. $I$ is center incirle . $B_1 C1$ and $\omega$ intersects at $M$ and $N$ . Find the ratio of circumradius of $ABC$ to circumradius $MIN$ .

$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.

In triangle $ABC$ $\omega$ is incircle and $\omega_1$ , $\omega_2$ , $\omega_3$ is tangents to $\omega$ and two sides of $ABC$ . $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$ . Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$

$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$ , $az+cx=b$ , $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$