\section*{Problem 5 - 081235}

Given are a triangular pyramid and the sphere circumscribed around it. The following statements are made about this pyramid and this sphere:

(1) A base edge of the pyramid is as long as the diameter of the sphere.

(2) The lengths of the other two base edges are in the ratio \(3: 4\).

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

(4) All edges of the pyramid are pairwise equal in length.

(5) The base of the pyramid is a right-angled triangle.

(6) The height of the pyramid is as long as the radius of the sphere.

It is known that one of the above six statements is false and the remaining statements are true.

How long are the edges of the pyramid?
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Statements 1) and 4) cannot both be satisfied simultaneously, as there can be at most one edge with the length of the diameter of the sphere.

A pyramid with a right-angled base with sides \(3 ; 4 ; 5\) and a height of 2.5, where the apex lies above the midpoint of the hypotenuse, satisfies conditions 1), 2), 5), and 6).

This has a volume of \(V=\frac{1}{3} G h=\frac{1}{3} \frac{3 \cdot 4}{2} \cdot 2.5=5\). Scaling by a factor of 2 results in a pyramid with a volume of 40 and base edges of \(6 ; 8 ; 10\). The other edges are of equal length, as the apex lies directly above the center of the circumcircle of the base and have a length of \(5 \sqrt{2}\).

This can be easily seen from the isosceles right triangle formed by the hypotenuse and the apex.

\section*{Source anonymous}