Baltic Way 1997 · Problem 11

Geometry

On two parallel lines, the distinct points $A_{1}, A_{2}, A_{3}, \ldots$ respectively $B_{1}$ , $B_{2}, B_{3}, \ldots$ are marked in such a way that $\left|A_{i} A_{i+1}\right|=1$ and $\left|B_{i} B_{i+1}\right|=2$ for $i=1,2, \ldots$ (see Figure). Provided that $\angle A_{1} A_{2} B_{1}=\alpha$ , find the infinite sum $\angle A_{1} B_{1} A_{2}+\angle A_{2} B_{2} A_{3}+\angle A_{3} B_{3} A_{4}+\ldots$

Parallel-line configuration shown beside Problem 11.

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Angles and distances · Transformations

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Solution

Solution: Answer: $\pi-\alpha$ . Let $C_{1}, C_{2}, C_{3}, \ldots$ be points on the upper line such that $|C_{i}C_{i+1}|=1$ and $B_{i}=C_{2i}$ for each $i=1,2, \ldots$ (see Figure 2). Then for any $i=1,2, \ldots$ we have

\angle A_{i}B_{i}A_{i+1}=\angle A_{i}C_{2i}A_{i+1}=\angle A_{1}C_{i+1}A_{2}=\angle C_{i+1}A_{2}C_{i+2}

Hence

\begin{aligned} & \angle A_{1}B_{1}A_{2}+\angle A_{2}B_{2}A_{3}+\angle A_{3}B_{3}A_{4}+\ldots= \\ & \quad=\angle C_{2}A_{2}C_{3}+\angle C_{3}A_{2}C_{4}+\angle C_{4}A_{2}C_{5}+\ldots=\pi-\alpha. \end{aligned}

Diagram for the mathnet 00zq 1 of bw-1997-11. Figure 2

Contest context

Results from Baltic Way 1997

11 teams

Mean score: 3.6 / 5
Scores of 4 or 5: 7 / 11
Estonia: 3 / 5

Score distribution

All team scores

Team	Score
Poland	5 / 5
Germany	5 / 5
Estonia	3 / 5
Sweden	5 / 5
Denmark	5 / 5
Latvia	5 / 5
Finland	0 / 5
Norway	5 / 5
St. Petersburg	5 / 5
Iceland	0 / 5
Lithuania	2 / 5

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