## Task B-4.1.

In the set of natural numbers, solve the equation $5^{x}+5^{y}+5^{z}=18775$, where $x<y<z$. How many triangles have side lengths that are numbers, not necessarily distinct, from the set $\{x, y, z\}$?
## Solution.

Let's write the given equation in the following form:

$5^{x}\left(1+5^{y-x}+5^{z-x}\right)=5^{2} \cdot 751$.

Since $x<y<z$ are natural numbers, the only possibility is:

$5^{x}=5^{2}$ and $1+5^{y-x}+5^{z-x}=751$.

Then $x=2, 5^{y-2}+5^{z-2}=750$.

Similarly, from $5^{y-2}\left(1+5^{z-y}\right)=5^{3} \cdot 6$ it follows that $5^{y-2}=5^{3}$ and $1+5^{z-y}=6$, i.e., $y-2=3$, so $y=5$.

From $1+5^{z-5}=6$ it follows that $5^{z-5}=5=5^{1}$, so $z-5=1$, hence $z=6$.

Thus, the solution to the given equation in the set of natural numbers is $x=2, y=5, z=6$.

These numbers determine one scalene triangle, three equilateral triangles, and several isosceles triangles. The isosceles triangles could be those with sides $225,226,552$, 556, 662, 665, but only those that satisfy the triangle inequality are possible, and these are 552, $556,662,665$.

Therefore, there are a total of eight sought triangles.