Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }\end{eqnarray}
Does this system have a solution?
The idea is to use the Brouwer's fixed-point theorem. If we consider the continuous function $f:\mathbb{R}^3\to\mathbb{R}^3$ defined by $$f(x,y,z)=(e^{xyz}\sqrt{e^{2xyz}+1},\cos(x+y+z)\sqrt{e^{2xyz}+1},\sin(x+y+z) \sqrt{e^{2xyz}+1}),$$
then it is enough to find a subspace $A$ of $\mathbb{R}^3$, homeomorphic to $D^3$ such that $f(A)\subset A$. Then by the Brouwer's fixed-point theorem, the function will have a fixed point in $A$, the one which will be our solution. But finding a suitable $A$ hasn't been an easy task for me. Probably I have to modify the function $f$ a little but I'm out of ideas.