## Nice characterization side-lengths of a triangle

Find the greatest such that and implies that are the side-lengths of a triangle.

*Solution:* Note that if then for we have

and are not the side-lengths of a triangle. Therefore . We can assume without loss of generality that . Pick and suppose that are not the side-lengths of a triangle. Then and we obtain

We have used the fact that (this comes from the fact that the power function is convex).

Consider now the function . Then and we can see that for is strictly increasing. Therefore the above inequalities imply that

which is a contradiction. Therefore, any makes the statement true, so the greatest is .

An interesting challenge is to generalize this. One could try to modify the power in the RHS to an arbitrary power : the problem turns to: find the least such that implies that are the sides of a triangle. At first sight we see that this cannot be true, since the above inequality cannot imply that are sides of a triangle since if we scale with the same dilation parameter, the RHS converges to zero with an order greater than the LHS, so at one point the inequality will hold, no matter who the initial are. Still, one can reproduce the above machinery and use the example to see that , and then for each if the inequality holds we obtain for and that

but this is not anymore a contradiction for . One immediate way to fix this is to assume are positive integers.

We have thus proved that the greatest such that and implies that are the side-lengths of a triangle is .