In a certain isosceles triangle, the base is in the ratio of $1: 2$ to the leg. Transform this triangle into an isosceles trapezoid whose area and perimeter are equal to the area and perimeter of the triangle, and whose adjacent sides are in the ratio of $1: 2$. What are the angles of the isosceles trapezoid?
In the case of an isosceles triangle with base $a$, the smaller side of the isosceles $x$, and one of its angles $\alpha$, according to the problem:

$$
5 \alpha = 6 x
$$

and

$$
\frac{a^{2}}{4} \sqrt{15} = 2 x^{2} \sin \alpha
$$

from which

$$
\sin \alpha = 0.18 \sqrt{15}
$$

and thus

$$
\alpha = 44^{\circ} 11^{\prime} 45^{\prime \prime}
$$

the other angle of the isosceles is

$$
\beta = 135^{\circ} 48^{\prime} 15^{\prime \prime}
$$

(Júlia Krampera, Debrecen.)

The problem was also solved by: E. Ádámffy, I. Dömény, V. Erdős, R. Földes, H. Friedländer, B. Gunszt, A. Haar, Z. Harsányi, P. Heimlich, Gy. Jánosy, G. Kertész, J. Kiss, I. Kürti, P. Messer, M. Pám, M. Riesz, J. Rosenberg, I. Schöffer, Gy. Schuster, Gy. Schwarz, J. Sonnenfeld, Gy. Söpkéz, E. Tandlich, the 5th district high school mathematics circle.