What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
1. Let \(a, b, c\) be the roots of the polynomial \(x^3 - ux^2 + vx - w\). By Vieta's formulas, we have:
   \[
   a + b + c = u, \quad ab + bc + ca = v, \quad abc = w
   \]

2. For \(a, b, c\) to be the sides of a triangle, they must satisfy the triangle inequality:
   \[
   a + b > c, \quad b + c > a, \quad c + a > b
   \]

3. We can use the symmetric sums of the roots to express the conditions. The triangle inequalities can be rewritten as:
   \[
   u - c > c, \quad u - a > a, \quad u - b > b
   \]
   Simplifying these, we get:
   \[
   u > 2c, \quad u > 2a, \quad u > 2b
   \]

4. Next, we use the identity for the sum of squares of the roots:
   \[
   a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) = u^2 - 2v
   \]

5. We also use the identity for the sum of cubes of the roots:
   \[
   a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) + 3abc
   \]
   Substituting the known values, we get:
   \[
   a^3 + b^3 + c^3 = u(u^2 - 3v) + 3w
   \]

6. Now, we consider the product of the sums of pairs of roots:
   \[
   (a + b)(b + c)(c + a) = (a + b + c)(ab + bc + ca) - abc = uv - w
   \]

7. For the roots to form a triangle, the product of the sums of pairs of roots must be greater than the product of the roots:
   \[
   (a + b)(b + c)(c + a) > abc
   \]
   Substituting the known values, we get:
   \[
   uv - w > w
   \]
   Simplifying this inequality, we obtain:
   \[
   uv > 2w
   \]

Conclusion:
The condition that must be satisfied by the coefficients \(u, v, w\) for the roots of the polynomial \(x^3 - ux^2 + vx - w\) to be the sides of a triangle is:
\[
\boxed{uv > 2w}
\]