$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)=f(x), g_{n+1}(x)=f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}=99$, find $ r_{99}$.

$ 2n+1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x+y)=f(x)f(y)-f(xy)+1$ for every $x,y\in\mathbb{Q}$.

$ c$ is a positive integer. Consider the following recursive sequence: $ a_1=c, a_{n+1}=ca_{n}+\sqrt{(c^2-1)(a_n^2-1)}$, for all $ n \in N$. Prove that all the terms of the sequence are positive integers.

The function $ f(x,y,z)=\frac{x^2+y^2+z^2}{x+y+z}$ is defined for every $ x,y,z \in R$ whose sum is not 0. Find a point $ (x_0,y_0,z_0)$ such that $ 0 < x_0^2+y_0^2+z_0^2 < \frac{1}{1999}$ and $ 1.999 < f(x_0,y_0,z_0) < 2$.

In a multiple-choice test, there are 4 problems, each having 3 possible answers. In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?