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Prove that the segments do not intersect for each k, n

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September 21, 2010
geometrypoint seteuclidean distanceIntersectionanglesIMO Shortlist

Problem Statement

Let A0,A1,…,AnA_0,A_1, \ldots , A_n be points in a plane such that (i) A0A1≤12A1A2≤⋯≤12n−1An−1AnA_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n and (ii) 0<∡A0A1A2<∡A1A2A3<⋯<∡An−2An−1An<180∘,0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ, where all these angles have the same orientation. Prove that the segments AkAk+1,AmAm+1A_kA_{k+1},A_mA_{m+1} do not intersect for each kk and nn such that 0≤k≤m−2<n−2.0 \leq k \leq m - 2 < n- 2.
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