To prove the existence of solution(s) of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$

Question

To prove if a certain equation has a solution, we do not need necessarily solve that equation.

Example: Is there any point of the curve $y=\sin^2(x)/x$ between $x=0$ and $x=\pi$ so that the slope of the tangent line at that point is zero?

The answer is yes, because $y$ is continuous, differentiable, and $y(0)=y(\pi)=0$ (limits take place). This is (Rolle's Theorem). And we are finish. We did not solve it and find where is that point.

My question is about a false conjecture that the equation

$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$$

has no positive integer solutions for $a,b,c$.

Without finding that/those solution(s), which is/are really large numbers, can we prove that the equation has a solution?

In other words, can we prove the existence of solution(s) but not by giving an examples like:

$a=154476802108746166441951315019919837485664325669565431700026634898253202035277999$

$b=36875131794129999827197811565225474825492979968971970996283137471637224634055579$

$c=4373612677928697257861252602371390152816537558161613618621437993378423467772036$

The proof can be by contradiction, or direct proof, or any way.

Thanks in advance