DE-BUNDESWETTBEWERB 2025

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.

Determine all real numbers $t$ with $0<t<1$ for which the following statement is true: If, in an acute-angled triangle, the sides with lengths $a$ , $b$ , and $c$ are opposite the angles with sizes $\alpha$ , $\beta$ , and $\gamma$ , as usual, and $t=\frac{a}{b+c}$ holds, then $\alpha \le \beta$ and $\alpha \le \gamma$ . Note: The correctness of the answer must be proven.

Two circles $k_1$ with center $M_1$ and $k_2$ with center $M_2$ intersect such that their two intersection points and their two centers are four vertices of a square. A point $P$ lies on $k_1$ , and two other distinct points $Q$ and $R$ lie on $k_2$ such that the segments $PQ$ and $PR$ have the same length as the segment $M_1M_2$ and $P$ and $R$ lie on the same side of the line $M_1M_2$ . Let the midpoints of the segments $PQ$ and $PR$ be denoted by $S$ and $T$ , respectively. Prove: The triangles $\triangle PSM_1$ and $\triangle PTM_2$ are similar.