Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a+b+c &=& 24\\ a^{2}+b^{2}+c^{2}&=& 210\\ abc &=& 440\end{array}\]

Let $ P$ be a point in the interior of the equilateral triangle $ \triangle{}ABC$ such that $ PA = 5$, $ PB = 7$, $ PC = 8$. Find the length of the side of the triangle $ ABC$.

Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}-ax^{3}+bx^{2}-cx+5 = 0$, knowing that they are real, positive and that: \[ \frac{r_{1}}{2}+\frac{r_{2}}{4}+\frac{r_{3}}{5}+\frac{r_{4}}{8}= 1.\]

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz-x^{2}}{1-x}=\frac{xz-y^{2}}{1-y}\] show that both fractions are equal to $ x+y+z$.

To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied: (1) $ f(rs) = f(r)+f(s)$ (2) $ f(n) = 0$, if the first digit (from right to left) of $ n$ is 3. (3) $ f(10) = 0$. Find $f(1985)$. Justify your answer.

Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that: \[\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}=\frac{2}{R}\]