\[180^\circ (n-2) = \sum_i \sum_j A_iA_jA_{i+1} \ge \sum_i \alpha_i (n-2) \implies 180^\circ \ge \sum_i \alpha_i. \]Equality occurs iff $A_1A_2...A_n$ cyclic.

Instantly solved, well done.

This problem is proposed by Gian Sanjaya (BlazingMuddy). My solution is the same as @2above so I won't post it

@above why spoil so early Here is the first official solution; the second one is the above solution. Consider an arbitrary $i = 3, 4, \ldots, n$, and notice that: 1. $\alpha_1 \leq \angle A_1 A_i A_2$; 2. $\alpha_j \leq \angle A_j A_1 A_{j + 1}$ for each $j = 2, 3, \ldots, i - 1$; and 3. $\alpha_j \leq \angle A_j A_2 A_{j + 1}$ for each $j = i, i + 1, \ldots, n$. Summing all the inequalities yield \[ \alpha_1 + \alpha_2 + \ldots + \alpha_n \leq \angle A_1 A_i A_2 + \angle A_2 A_1 A_i + \angle A_1 A_2 A_i = 180^{\circ}. \]If equality holds, then the above inequality are all equalities. In particular, $\angle A_1 A_i A_2 = \alpha_1 = \angle A_1 A_j A_2$ for each $3 \leq i, j \leq n$. This forces $A_1 A_2 \ldots A_n$ to be cyclic. Conversely, if it is cyclic, it is easy to see that $\alpha_i = \angle A_i A_j A_{i + 1}$ for any $j \neq i, i + 1$, and thus the desired equality holds. Maybe godjuansan wants to post a barycentric solution?

No geometry problems in day 2 hmm

Firefly123 wrote: No geometry problems in day 2 hmm I think this was classified as geometry

BR1F1SZ wrote: Firefly123 wrote: No geometry problems in day 2 hmm I think this was classified as geometry I guess so, as a geometry hater, this problem really help boost my score

somebodyyouusedtoknow wrote: Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \]and determine all equality cases. Short and elegant problem, i wonder if its inspired from IMO problem

Does this inequality still hold for any concave polygon?