378. The Pancake Problem. A cook is baking pancakes on a round skillet, the diameter of which is 26 units. She places three round pieces of dough of different sizes such that their centers lie on the same straight line and together they cover the entire diameter of the skillet, but - only half of its area. Find the diameters of the three pancakes, given that they are expressed as whole numbers.
378. Since the areas are proportional to the squares of the corresponding diameters, it is sufficient to find 3 different integers, the sum of which is 26 and the sum of whose squares is $676: 2=338$.

Let $x, y, z$ be the diameters of the pancakes. Then

$$
\begin{gathered}
x+y+z=26 \\
x^{2}+y^{2}+z^{2}=338
\end{gathered}
$$

and, therefore,

$$
x^{2}+y^{2}+z^{2}=\frac{(x+y+z)^{2}}{2}
$$

This leads to the Diophantine equation:

$$
x^{2}+y^{2}+z^{2}-2 x y-2 y z=0
$$

or

$$
(x-y-z)^{2}=4 y z
$$

Thus, $y z$ is a perfect square, so $y=k^{2} m$, $z=l^{2} m$ for some integers $k, l, m$ and

$$
\begin{gathered}
x-y-z= \pm 2 k l m \\
x=m(k \pm l)^{2}
\end{gathered}
$$

Therefore,

$$
\begin{aligned}
2 \cdot 13=26 & =m\left[(k \pm l)^{2}+k^{2}+l^{2}\right] \\
2 \cdot 13^{2}=338 & =m^{2}\left[(k \pm l)^{4}+k^{4}+l^{4}\right]
\end{aligned}
$$

From this, $m=1$ or 13. Direct verification shows that the latter is impossible. It remains to find three different squares that sum to 26. The only numbers that satisfy this condition are $1, 9, 16$.

[A. Brusso, M. M., 41, 100 (February 1968).]