\color{red}{9} distinct lines extend infinitely in the two-dimensional Euclidean plane. Prove that there exists an angle formed by \color{cyan}{\text{two}} of them with a degree of at most \color{red}{20}.
Moreover, show that this bound is the best possible.
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Let \alpha_1,\alpha_2,\dots,\alpha_8 be the positive directed angles between the first line and the other lines. WLOG assume \alpha_1 < \alpha_2 < \dots < \alpha_8. now notice that if \exists i, 20^\circ\ge\alpha_{i+1}-\alpha_{i} then we can get the angle between the two lines and it wil be the difference. which means that 20^\circ\ge\alpha_{i+1}-\alpha_{i} \forall i \Rightarrow \alpha_8 > 20\cdot 8 = 160^\circ \Rightarrow the acute angle opposite to \alpha_8 is less than 20.
this is of course the best bound because of the example where you have a regular 9-gon
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