HMMT November, 2017

Solution:
Substitute $x = \cos \theta$ and $\sqrt{1-x^{2}} = \sin \theta$, and notice that $58 = 3^{2} + 7^{2}$, $42 = 2 \cdot 3 \cdot 7$, $149 = 7^{2} + 10^{2}$, and $140 = 2 \cdot 7 \cdot 10$. Therefore the first term is an application of Law of Cosines on a triangle that has two sides $3$ and $7$ with an angle measuring $\theta$ between them to find the length of the third side; similarly, the second is for a triangle with two sides $7$ and $10$ that have an angle measuring $90^{\circ} - \theta$ between them. "Gluing" these two triangles together along their sides of length $7$ so that the merged triangles form a right angle, we see that the minimum length of the sum of their third sides occurs when the glued triangles form a right triangle. This right triangle has legs of length $3$ and $10$, so its hypotenuse has length $\sqrt{109}$.