Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$(variation but same answer) Answer3

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules: $1$. It must be in the interior of the island (none on borders). $2$. No two airports can be at the exact same location. $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle. $4$. No three airports can be collinear. Find the maximum value of $k$ for each $n$ Harder VersionReplace $1$ new and $2$ old with $1$ old and $2$ new.