Problems

Let $a$, $b$ and $c$ be the three side lengths of a triangle. Does there exist a triangle with side lengths $a+1$, $b+1$ and $c+1$? Does it depend on what $a$, $b$ and $c$ are?

There is a triangle with side lengths $a$, $b$ and $c$. Can you form a triangle with side lengths $\frac{a}{b}$, $\frac{b}{c}$ and $\frac{c}{a}$? Does it depend on what $a$, $b$ and $c$ are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on $a$, $b$ and $c$.
Recall that a triangle can be drawn with side lengths $x$, $y$ and $z$ if and only if $x+y>z$, $y+z>x$ and $z+x>y$.

There is a triangle with side lengths $a$, $b$ and $c$. Does there exist a triangle with side lengths $|a-b|$, $|b-c|$ and $|c-a|$? Does it depend on what $a$, $b$ and $c$ are?
Recall that a triangle can be formed with side lengths $x$, $y$ and $z$ if and only if all the inequalities $x+y>z$, $y+z>x$ and $z+x>y$ hold.

There is a triangle with side lenghts $a$, $b$ and $c$. Does there exist a triangle with sides of lengths $a^2+bc$, $b^2+ca$ and $c^2+ab$? Does it depend on the values of $a$, $b$ and $c$?