BdMO National Secondary 2020 P2

[Mursalin]

একটি বহুভুজকে 'সুন্দর বহুভুজ' বলা যাবে, যদি তার তিনটি শীর্ষ বেছে নেওয়া যায় যেন তাদের মাঝে \(144^\circ\) কোণ তৈরী হয়। \(n\) বাহু বিশিষ্ট এমন কতগুলো সুষম বহুভুজ আছে যাদেরকে সুন্দর বহুভুজ বলা যাবে যেখানে \(8<n\leq 2024\)?


A polygon is called beautiful if you can pick three of its vertices to have an angle of \(144^\circ\). Compute the number of integers \(8<n\leq 2024\) for which a regular \(n\)-gon is beautiful.

[Mehrab4226]

একটি বহুভুজকে 'সুন্দর বহুভুজ' বলা যাবে, যদি তার তিনটি শীর্ষ বেছে নেওয়া যায় যেন তাদের মাঝে \(144^\circ\) কোণ তৈরী হয়। \(n\) বাহু বিশিষ্ট এমন কতগুলো সুষম বহুভুজ আছে যাদেরকে সুন্দর বহুভুজ বলা যাবে যেখানে \(8<n\leq 2024\)?


A polygon is called beautiful if you can pick three of its vertices to have an angle of \(144^\circ\). Compute the number of integers \(8<n\leq 2024\) for which a regular \(n\)-gon is beautiful.

At first, I will try to make the question a little bit easier to digest. Any selecting any three points and an angle formed is just the angles formed by the sides and diagonals of the polygon. Since we are only concerned with regular polygons, drawing the circumcircle can help us a lot.

We draw the circumcircle of the polygon, the circle is divided in n equal arcs by the n-gon. Now the angles we are concerned with are formed on any vertex of the polygon in other words the on the circle. Now Each equal arc subtends an angle of $\frac{180}{n}$ degrees. Now depending on where we choose our points, we can use the unitary method to find the angle between them.

For example, let us find the equal angles(the angle between adjacent sides) of the polygon, We can take any vertex, and then we can clearly see that the angle subtended by $(n-2)$ arcs. Thus the angles are $\frac{(n-2)\times 180}{n}$ similarly the other angles are formed, where each angle is subtended by $(n-k)$ arcs, where $1< k < n$.

Now if let n = 2024, and n-gon is good, let us find the value of k,
$\frac{180(2024-k)}{2040} = 144 $
$\therefore k =404.8$

Now all values from 2 to 404 must have a value of $n$ in between the range such that the polygon is good or in other words
$\frac{180(n-k)}{n} = 144 $ for all those values.
Therefore the required number =v(404-2)+1=403
I hope my solution is correct.