We received the following statements about a triangular pyramid and its circumscribed sphere:

I. One base edge is equal in length to the diameter of the sphere.

II. The ratio of the lengths of the other two base edges is $3: 4$.

III. The volume of the pyramid is $40 \mathrm{~cm}^{3}$.

IV. The edges of the pyramid are equal.

V. The base of the pyramid is a right-angled triangle.

VI. The height of the pyramid is equal to the radius of the sphere.

Furthermore, it was communicated that one of these six statements is false, while the others are true.

What are the lengths of the pyramid's edges?
The IV. statement is false because it contradicts both the II. and the I., since from every point on the sphere, only one diameter-length chord can originate; from each vertex of the tetrahedron, however, 3 edges originate. Thus, if the IV. were true, there would be more than one false statement. (The V. would also be false, see the remark.)

After this - denoting the radius of the sphere by $r$ - the sides of the base triangle are $2 r, 8 r / 5, 6 r / 5$ according to I. and II.; the area of the base triangle is $(8 r / 5) \cdot (6 r / 5) / 2$ according to V.; its height is $r$ according to VI., so according to III.

$$
\frac{1}{2} \cdot \frac{8 r}{5} \cdot \frac{6 r}{5} \cdot \frac{r}{3}=40, \quad \text { from which } \quad r=5 \text{ cm}
$$

thus the base edges are $10, 8, 6 \text{ cm}$.

The base triangle lies in the plane of a great circle of the sphere, so since $m=r$, the fourth vertex is the endpoint of the diameter perpendicular to the great circle, hence its projection onto the base is the center of the sphere and the circumcircle of the base. Thus, the 3 side edges are equal, their common length is $r \sqrt{2} \approx 7.07 \text{ cm}$.

Gyula Pekár (Székesfehérvár, József A. Gymn., III. grade)
![](https://cdn.mathpix.com/cropped/2024_05_02_69ff32c8b520e5c83d3fg-1.jpg?height=438&width=900&top_left_y=740&top_left_x=602)

Remark. Every plane section of the sphere containing a diameter is a great circle, and the diameter of the sphere is also the diameter of this great circle, so V. follows from I. by Thales' theorem, hence it is redundant. On the other hand, however, the legs of a right triangle are shorter than the hypotenuse, so IV. also contradicts V.