$\omega$ 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 $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]