XK-TST 2016

Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .

If quadratic equations $x^2+ax+b=0$ and $x^2+px+q=0$ share one similar root then find quadratic equation for which has roots of other roots of both quadratic equations .

It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied $f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$ If $g(x)=f(x)-x+1$ then find $g(2016)$ .

Let be $ABC$ an acute triangle with $|AB|>|AC|$ . Let be $D$ point in side $AB$ such that $\angle ACD=\angle CBD$ . Let be $E$ the midpoint of segment $BD$ and $S$ let be the circumcenter of triangle $BCD$ . Show that points $A,E,S$ and $C$ lie on a circle .