Sergeev I.N.

At the base $A_{1} A_{2} \ldots A_{n}$ of the pyramid $S A_{1} A_{2} \ldots A_{n}$ lies a point $O$, such that $S A_{1}=S A_{2}=\ldots=S A_{n}$ and $\angle S A_{1} O=\angle S A_{2} O=$ $\ldots=\angle S A_{n} O$.

For what least value of $n$ does it follow from this that $S O$ is the height of the pyramid?
By the Law of Sines for triangles $S A_{k} O (k=1,2, \ldots, n)$, we have $\sin \angle S O A_{k} = \frac{S A_{k}}{S O} \cdot \sin \angle S A_{k} O$.

Since the right-hand side of this equality does not depend on the choice of $k=1,2, \ldots, n$, the value of $\sin \angle S O A_{k}$ also does not depend on this choice. Therefore, for different choices of $k$, the angle $S O A_{k}$ can take no more than two different values, each of which, together with the lengths $S O, S A_{k}$, and the angle $S O A_{k}$, uniquely determines the triangle $S A O$.

If $n \geq 5$, then among the triangles $S A_{k} O (k=1,2, \ldots, 5)$, there are at least three identical ones. Let these, for definiteness, be the triangles $S A_{1} O, S A_{2} O$, and $S A_{3} O$. Since $O A_{1} = O A_{2} = O A_{3}$, the point $O$ is the center of the circumscribed circle of the triangle $A_{1} A_{2} A_{3}$. Let $S H$ be the height of the pyramid $S A_{1} A_{2} A_{3}$. Then the point $H$ is also the center of the circumscribed circle around $A_{1} A_{2} A_{3}$, i.e., $H = O$.

For $n=4$, it does not follow from the condition that $S O$ is the height of the pyramid. For example, if $A_{1} A_{2} A_{3} A_{4}$ is an isosceles trapezoid, $O$ is the intersection point of its diagonals, and $H$ is the center of the circumscribed circle around it, then for the pyramid $S A_{1} A_{2} A_{3} A_{4}$, where $S H$ is the height to the base, all the conditions of the problem are satisfied. Indeed, first, $S A_{1} = S A_{2} = S A_{3} = S A_{4}$ due to the equality (by two legs) of triangles $S H A_{1}, S H A_{2}, S H A_{3}$, and $S H A_{4}$, and secondly,

$\angle S A_{1} O = \angle S A_{2} O = \angle S A_{3} O = \angle S A_{4} O$ due to the equality (by three sides) of isosceles triangles $S A_{1} A_{3}$ and $S A_{2} A_{4}$. In this case, $H \neq O$. For $n \leq 4$, it also does not follow from the condition that $S O$ is the height of the pyramid (the corresponding example is obtained from the previous one by considering its part - the pyramid $S A_{1} A_{2} A_{3}$).

![](https://cdn.mathpix.com/cropped/2024_05_06_5ff42d6d57f9dccbb1f5g-44.jpg?height=515&width=500&top_left_y=0&top_left_x=779)

## Answer

For $n=5$.