A3. Sara is looking for a three-digit natural number $\overline{x y z}$ (where $z$ are the units, $y$ are the tens, and $x$ are the hundreds), for which $1 \leq x<y<z$ holds, and the sum of the numbers $\overline{x y z}, \overline{y z x}$, and $\overline{z x y}$ is a three-digit number with all digits being the same. What is the maximum number of such three-digit numbers that Sara can find?
(A) 3
(B) 5
(C) 7
(D) 9
(E) 11
A3. The sum of the numbers $\overline{x y z}, \overline{y z x}$ and $\overline{z x y}$ is equal to $100(x+y+z)+10(y+z+x)+(z+x+y)$. Since this must be a three-digit number, $x+y+z<10$ and in this case, all digits of this number are automatically equal. We are therefore looking for the number of triples of numbers $(x, y, z)$ for which $1 \leq x<y<z$ and $x+y+z<10$. There are exactly 7 such triples, which are $(1,2,3),(1,2,4),(1,2,5)$, $(1,2,6),(1,3,4),(1,3,5)$ and $(2,3,4)$. Sara has 7 such three-digit numbers to choose from.