## Task 1 - 200511

Ralph, an avid reader of the mathematical student magazine alpha, presents the following problem to his classmates in a study group:

In the illustrated figure, natural numbers are to be entered for $a, h, l, p$ such that the sum in each of the two diagonals $D_{1}, D_{2}$ is

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0087.jpg?height=274&width=402&top_left_y=451&top_left_x=1358)
135. The number $p$ should be three times the number $a$, and the number $h$ should be five times the number $l$.

Determine all such entries, and explain how to find them! Also, verify that all the required conditions are met!
If an entry of natural numbers for $a, h, l, p$ satisfies the conditions, then $p=3a$. In the diagonal ${ }_{D} 1$, the sum is thus $a+3a+a=5a$. Therefore, $5a=135$, so $a=135: 5=27$, and since $p=3a$, $p=3 \cdot 27=81$. Furthermore, $h=5l$.

Thus, in the diagonal $D_{2}$, the sum is $l+81+5l=6l+81$. Therefore, $6l+81=135$ and hence $6l=135-81=54$, so $l=54: 6=9$, and since $h=5l$, $h=5 \cdot 9=45$. Therefore, only the entry

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0087.jpg?height=197&width=209&top_left_y=1192&top_left_x=929)

can satisfy all the required conditions.

#### Abstract