Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$.

It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic. Find the ratio $\tfrac{HP}{HA}$.

It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$. a) Find the greatest value of $a$, such that this equation has at least one real root. b) Find all the values of $a$, such that the equation has at least one real root.

Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that \[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]

Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: \[ x,y > 0 \qquad f(x+f(y)) = yf(xy+1). \] a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$. b) Find all such functions that require the given condition.