Prove $\, \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\in\Bbb Z \,\Rightarrow\, \frac{ab}{c},\frac{ac}{b},\frac{bc}{a}\in\Bbb Z $
Problem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a} $ is integer.
(Proposed by Gerhard J. Woeginger)
The solution is as follows: Set $u := ab/c, v := ac/b$ and $w := bc/a$. By assumption, $u + v + w$ is an integer. It is easily seen that $uv + uw + vw = a^2 + b^2 + c^2$ and $uvw = abc$ are integers, too. According to Vieta’s formulæ, the rational numbers $u, v, w$ are the roots of a cubic polynomial $x^3 + px^2 + qx + r$ with integer coefficients. As the leading coefficient is 1, these roots are integers.
Question: I haven't found anything supporting this, but why does a cubic equation with integer coefficients and a leading coefficient of 1 have integer roots? Why does the cubic equation $x^3+5x^2-20x+5=0$ not have integer roots then?
It follows from the rational root theorem that every rational root of every monic polynomial (of any degree) with integer coefficients must be integer.
Besides, the polynomial $x^3+5x^2-20x+5$ has no rational roots.