A regular triangular pyramid has a base edge of unit length, and its side faces form $120^{\circ}$ angles with each other. What are the lengths of the pyramid's side edges?
Let $A, B, C, D$ be the vertices of the pyramid, with $ABC$ being the base triangle, which is known to be equilateral, and $D$ being the fourth vertex. The dihedral angle between the lateral faces is measured by the angle between the perpendiculars to the line of intersection of the two planes at a point on that line. Draw perpendiculars from $A$ and $B$ to the edge $CD$; since the pyramid is regular, the plane through the midpoint of $AB$ and the edge $CD$ is a plane of symmetry of the pyramid, and thus the two perpendiculars intersect the edge $CD$ at a common point $E$. The angle $\angle AEB = 120^\circ$.

Draw the altitude $EF$, and note that $\triangle AEF$ is half of an equilateral triangle, whose altitude is $\frac{1}{2}$. Since we know that in an equilateral triangle, the altitude is $\frac{\sqrt{3}}{2}$ times the side length, we can write $\frac{1}{2} = \frac{a\sqrt{3}}{2}$, from which $AE = a = \frac{\sqrt{3}}{3}$. Knowing this, in the right triangle $\triangle ACE$,

$$
EC = \sqrt{1^2 - \left(\frac{\sqrt{3}}{3}\right)^2} = \frac{\sqrt{6}}{3}
$$

There are several ways to calculate the length of the edge $CD$ (e.g., similarity, Pythagorean theorem, etc.).

The triangle $\triangle ADC$ is isosceles (since the pyramid is regular). Let $x$ be the length of the edge $CD$, and let the perpendicular projection of $D$ onto $AC$ be $D'$, and the projection of $E$ onto $AC$ be $E'$.

Since $AE' = \frac{\sqrt{3}}{3} < EC = \frac{\sqrt{6}}{3}$, $E'$ is closer to $A$ than $D'$, which is the midpoint of $AC$, so $E$ is farther from $C$ than $D$. In the triangle $\triangle ADE$, we can apply the Pythagorean theorem:

$$
\left(x - \frac{\sqrt{6}}{3}\right)^2 + \left(\frac{\sqrt{3}}{3}\right)^2 = x^2
$$

from which the length of the pyramid's edge $x = \frac{\sqrt{6}}{4} \approx 0.6123$ units.

Remarks. The majority of the incomplete solutions resulted from the fact that many only drew diagrams and performed calculations but failed to justify why they calculated what they did. Very few wrote down the definition of the dihedral angle between two planes, or why in our case the two perpendiculars intersect at a point.

Many used the cosine rule or trigonometric functions. Naturally, if their solution was otherwise correct, they also received full marks. However, in the solutions, we should strive for simplicity, and where possible, use elementary methods to avoid complicating the calculations, as this can lead to more errors.