A regular pyramid's base and side face inclination angle is equal to the inclination angle between the side edge and base edge originating from one vertex. What is this angle?
A regular pyramid has a regular polygon as its base, and its side faces are congruent isosceles triangles.

Let $A$ and $B$ be two adjacent vertices of the base, $F$ the midpoint of the edge $AB$, and $C$ the vertex of the pyramid not on the base. Since the pyramid is regular, the orthogonal projection of $C$ onto the base is exactly the center $O$ of the circle circumscribed around the base.

$1986-05-214-2 . \mathrm{eps}$

The lines $OF$ and $CF$ are both perpendicular to the edge $AB$, so the angle $\angle OFC$ is precisely the dihedral angle between the base and a side face of the pyramid. The angle $\angle FAC$ is the angle between the side edge and the base edge starting from vertex $A$, so according to the condition:

$$
\angle OFC = \angle FAC.
$$

The triangles $COF$ and $CFA$ are right-angled, and due to (1) they have one more equal angle. Therefore, these two triangles are similar, and the ratios of corresponding sides are equal:

$$
\frac{CO}{OF} = \frac{CF}{FA}
$$

or

$$
\frac{CO}{CF} = \frac{OF}{FA}
$$

Since $CO > OF$ because in a triangle, the larger side is opposite the larger angle. This implies that

$$
\angle AOF > 45^{\circ} \text{ so } \angle AOB = 2 \cdot \angle AOF > 90^{\circ}.
$$

However, the only regular polygon where the side appears at an angle greater than $90^{\circ}$ from the center is the regular triangle. Therefore, the base of the pyramid is a regular triangle.

Then $\angle AOB = 120^{\circ}$, so $\angle FOA = 60^{\circ}$, and thus $\angle FAO = 30^{\circ}$. In the right-angled triangle $OFA$,

$$
\frac{OF}{FA} = \tan 30^{\circ} = \frac{\sqrt{3}}{3}
$$

and from (2) we get

$$
\frac{CO}{CF} = \frac{\sqrt{3}}{3}
$$

This is precisely the sine of the desired angle. From this, the desired angle is $35^{\circ} 15'$.

Since the obtained angle is acute, such a regular pyramid indeed exists. This completes the solution of the problem.

Note. Many submissions interpreted the regular pyramid as having all edges equal. These solutions did not receive points.