[ Non-Convex Polygons]

What is the greatest number of acute angles in a non-convex $n$-gon?

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Let $k$ be the number of acute angles in an $n$-sided polygon. Then the sum of its angles is less than $k \cdot 90^{\circ} + (n-k) \cdot 360^{\circ}$. On the other hand, the sum of the angles of an $n$-sided polygon is $(n-2) \cdot 180^{\circ}$ (see problem 22.23), so $k \cdot 90^{\circ} + (n-k) \cdot 360^{\circ} > (n-2) \cdot 180^{\circ}$, i.e., $3k < 2n + 4$. Therefore, $k \leq [2n / 3] + 1$, where $[x]$ denotes the greatest integer not exceeding $x$.

Examples of $n$-sided polygons with $[2n/3] + 1$ acute angles are shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_9505f60a586b58f27d0ag-13.jpg?height=489&width=480&top_left_y=207&top_left_x=183)

$n=3m$

![](https://cdn.mathpix.com/cropped/2024_05_06_9505f60a586b58f27d0ag-13.jpg?height=480&width=460&top_left_y=209&top_left_x=799)

$n=3m+1$

![](https://cdn.mathpix.com/cropped/2024_05_06_9505f60a586b58f27d0ag-13.jpg?height=495&width=461&top_left_y=204&top_left_x=1416)

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