## Task 1 - 160511

In a mathematics working group, Monika presents the participants with the following problem:

Each of the letters $A, L, P, H$ represents a single-digit natural number. The following conditions apply:

(1) The number $H$ is twice as large as the number $P$.

(2) The number $A$ is equal to the sum of the number $P$ and twice the number $H$.

(3) The number $L$ is equal to the sum of the numbers $A, P$, and $H$.

If the numbers $A L P H A$ are written in this order, one obtains the (five-digit) reader number of the mathematical student magazine "alpha".

What is this reader number?
As equations, we get

$$
H=2 P ; \quad A=P+2 H ; \quad L=A+P+H
$$

Additionally, $A, L, P, H$ are digits from 0 to 9.

By substituting the first equation into the second and third, we get $A=5 P$ and $L=A+3 P$, and consequently $L=8 P$. Since $P$ and $L$ are less than 10, $P$ must be 1 and $L$ must be 8, and thus $H=2, A=5$.

Therefore, "alpha" has 58125 readers.