Baltic Way 2021 · Shortlist problem

Geometry

Let's call the intersection of two segments almost perfect if for each of them the length of the segment is at least $2021$ times the distance between its midpoint and the intersection point. Prove that there exists a closed broken line that intersects each of its segments at least once and for which all its intersections are almost perfect.

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Topics

Constructions, loci, concurrency and collinearity · Transformations · Triangles and centers

Solutions

Solution

Consider two equilateral triangles with common centre and parallel sides. The closed broken line $A_1B_2C_1A_2B_1C_2$ has three intersections, because of symmetry we will consider only one. Assume that $A_1B_2$ intersects $A_2B_1$ in point $X$ . Triangles $A_1B_1X$ and $A_2B_2X$ are similar therefore $\frac{A_1X}{XB_2} = \frac{A_1B_1}{A_2B_2}$ . It is enough to choose initial triangles of close enough size $1 < \frac{A_1B_1}{A_2B_2} < \frac{2022}{2021}$ to make the intersection $X$ almost perfect. $\square$