In a certain number system, the square of 34 in the same system is 1552. What is the base number of this system?
If the base number of the number system is $x$, then according to our data,

$$
(3 x+4)^{2}=x^{3}+5 x^{2}+5 x+2
$$

After performing the indicated operation and simplifying:

$$
x^{3}-4 x^{2}-19 x=x\left(x^{2}-4 x-19\right)=14
$$

Thus, $x$ is a divisor of 14, but this divisor must be greater than 5, because the base number must be greater than the digits of 1552. Therefore, $x$ is either 7 or 14. However, $x$ is less than 10, because in the decimal system $34^{2}=1156$. $x=7$ satisfies the equation

$$
7\left(7^{2}-4 \cdot 7-19\right)=7(49-47)=14
$$

The sought base number is:

$$
x=7
$$

In the decimal system:

$$
(3 \cdot 7+4)^{2}=25^{2}=625 \quad \text { and } \quad 7^{3}+5 \cdot 7^{2}+5 \cdot 7+2=625
$$

Györke Zsolt (state gymnasium VI. grade, Beregszász).