DE-BUNDESWETTBEWERB 2025 Problem 5

The current year $2025$ has the following special property: Its decimal representation is created by writing the two numbers $20$ and $25$ one after the other, and $2025$ is also the square of the sum of these two numbers, i.e. $2025 = (20 + 25)^2$ . We call a positive integer $q$ $k$ -curious ( $k \ge 1$ ) if it has a $2k$ -digit decimal representation with the following property: The first $k$ digits and the last $k$ digits each form a decimal representation of a number $x$ or $y$ , respectively, and the following applies: $q=(x+y)^2$ . a) Prove: If for any $k$ there exists at least one $k$ -odd number divisible by $5$ , then there also exists at least one $k$ -odd number divisible by $2025$ . b) Are there infinitely many $k$ for which there exists at least one $k$ -odd number divisible by $2025$ ?

Notes: For the purposes of this question, decimal representations of numbers may contain leading zeros. The correctness of the answer must be proven.