# German contest "Bundeswettbewerb Mathematik" $2019$ [closed]

Recently this question was asked. This is exactly task $$4.$$ of the German mathematical contest "Bundeswettbewerb Mathematik". Since it is forbidden to ask questions of an on-going contest here on MSE I flagged the question as such one. Furthermore I will add within this post all four tasks translated into English in order to prevent such an unfair behavior until the contest has come to an end $$($$which will be at the fourth of march $$2019)$$. For those who are capable of understanding German by themselves: you can find the tasks at the official website.

Task $$\textbf{1}$$
A $$8$$x$$8$$ chess board is completely and without overlapping covered by $$32$$ $$1$$x$$2$$ dominoes.

Prove: There are always two dominoes which form a $$2$$x$$2$$ square.

Task $$\textbf{2}$$
The letters $$A,C,F,H,L,$$ and $$S$$ are six not necessarily distinct digits in the decimal system with $$S\ne0$$ and $$F\ne 0$$. From them the two six-digit decimal numbers $$SCHLAF$$ and $$FLACHS$$ are formed.

Prove: The difference of these two numbers is divisible by $$271$$ if and only if $$C=L$$ and $$H=A$$.

Task $$\textbf{3}$$
Within the square $$ABCD$$ the point $$E$$ one the side $$BC$$ and the point $$F$$ on the side $$CD$$ are chosen such that $$\angle EAF = 45 °$$. Moreover both $$E$$ and $$F$$ are no vertices of the square. The two straight lines $$AE$$ and $$AF$$ intersect with the circumcircle of the square beside point $$A$$ within the two points $$G$$ and $$H$$ respectively.

Prove that $$EF$$ and $$GH$$ are parallel.

Task $$\textbf{4}$$
Within the decimal representation of $$\sqrt{2}=1.4142...$$ Isabelle finds a sequence of $$k$$ consecutive zeroes where $$k$$ is a positive integer.

Prove: The first zero of this sequence appears at the earliest at the $$k$$th decimal place after the comma.

EDIT
A quick research on the main site shows that it seems like — at least right now — there are no other question of the contest asked yet. Anyway I am not sure whether I searched in the right manner in order to identify possible questions of the contest. In the case that anyone actually finds one of these question feel free to add them within this post $$($$either as answer or as an edit$$)$$.

• Thanks for this!! Dec 1 '18 at 23:57
• @amWhy No problem. It makes me furious that the OP of the linked post claims that "It's from a competition of my country a few years ago". I marked this comment as In need of moderator intervention aswell since it is an obvious lie. I hope for the closure of this question hence it is such an unfair behavior towards all other honest contenders. Dec 2 '18 at 0:03
• Not concerning this situation in particular, but it does seem a bit extreme that a contests like this can prohibit discussion of a problem for, in this case, several months. Dec 2 '18 at 22:14
• @mrtaurho The user "Elsa" who asked the first offending question also asked this question, which is a thinly disguised version of task 3 of Bundeswettbewerb Mathematik. Dec 6 '18 at 10:05
• Task $3$ is not clear to me. $AE$ (resp. $AF$) intersects with the perimeter at $A$ and $E$ (resp. $A$ and $F$). What is $G$ (resp. $H$)?
– Aloizio Macedo Mod
Dec 6 '18 at 15:19
• @Aloizio Macedo The point where $AE$ (resp. $AF$) intersects with the perimeter of the square is defined as point $G$ (resp. $H$). The point $E$ (resp. $F$) lies on the edge $BC$ (resp. $CD$). Maybe the term 'perimeter' is misleading. The points $G$ and $H$ are elements of the circumference of a circle defined by the square $ABCD$ where all vertices lie on this circle. Dec 6 '18 at 16:56
• @mrtaurho Perfect. Thanks for the clarification.
– Aloizio Macedo Mod
Dec 6 '18 at 17:07
• @Aloizio Macedo I am not familiar with the English terms concerning geometry similiar to Task $3$. Therefore it is possible that I eventually misused the word "perimeter". In German the circle to which the task refers is a so-called "Umkreis" which can be roughly translated as "outer circle". Anyway I do believe that this is actually the proper English term so I tried to describe what is meant. Should I include the clarification by reformulating Task $3$ or is it fine with a further explanation within the comments alone? Dec 6 '18 at 17:12
• @mrtaurho I think that the more appropriate term would be "circumcircle". This is my guess based on the word that is used in my native tongue, so I could be wrong. (It matches your translation of "outer circle", though.) I will change the wording in your post.
– Aloizio Macedo Mod
Dec 6 '18 at 17:17
• If anyone feels that the term is not the proper one, feel free to change it back. (Or to something else.)
– Aloizio Macedo Mod
Dec 6 '18 at 17:18
• I saw task 1 somewhere recently too with an answer, possibly on Puzzling.SE Jan 3 '19 at 20:03
• The deadline of this contest was two days ago. The post is not longer needed as it was intended therefore I am voting to close this question. Mar 6 '19 at 9:46