I want to construct a triangle $ABC$ with sides $|AB|=3 \text{~cm}$ and $|BC|=4 \text{~cm}$. Furthermore, I want to construct all circles, each of which will have its center at one of the vertices of the triangle and will pass through another vertex.

How long must the side $AC$ be so that there are exactly five such circles? Determine all possibilities.

(V. Hucíková)
If sides $AB$ and $AC$ were of the same length, then the circle centered at point $A$ passing through point $B$ would also pass through point $C$. In such a case, Ema would construct only one circle centered at point $A$.

If sides $AB$ and $AC$ were of different lengths, then the circle centered at point $A$ passing through one of the points $B$ and $C$ would not pass through the other. In such a case, Ema would construct two circles centered at point $A$.

Similar cases can occur for circles centered at point $C$. Circles centered at point $B$ will certainly be two, because from the problem statement we know that sides $BA$ and $BC$ are of different lengths.

If the sides of triangle $ABC$ were all different, then Ema would construct $2+2+2=6$ circles. If side $AC$ were equal to one of the remaining two sides, then Ema would construct $1+2+2=5$ circles. Therefore, side $AC$ must be either $3 \text{ cm}$ or $4 \text{ cm}$.
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Scoring suggestion. 1 point for each of the two valid options; 3 points for the correct justification of the number of circles; 1 point for explaining that no other options are possible.

Justification of the correct number of circles for only one of the two valid options is considered sufficient.