## Task 2 - 180722

A fraction is required to have the following two properties (1), (2). Determine all fractions that meet this requirement!

(1) The fraction represents the same fractional number as 0.4.

(2) The sum of the numerator and the denominator of this fraction is a two-digit square number.
Assume there is such a fraction $\frac{p}{q}$ with natural numbers $p, q$ and $q \neq 0$. Then, due to (1), it follows that $\frac{p}{q}=\frac{2}{5}$.

This implies $p=2n$ and $q=5n$ (with $n \in \mathbb{N}, n \neq 0$), so $p+q=7n$, which is equivalent to $7 \mid p+q$.

Since 49 is the only two-digit square number divisible by 7, due to (2), it must be that $p+q=49$ and thus $n=7, p=14, q=35$. Therefore, if there is a fraction with the required properties, it can only be the fraction $\frac{14}{35}$. Indeed, this fraction satisfies both conditions.