Let $ABC$ be an acute triangle. Let$H$ and $D$ be points on $[AC]$ and $[BC]$, respectively, such that $BH \perp AC$ and $HD \perp BC$. Let $O_1$ be the circumcenter of $\triangle ABH$, and $O_2$ be the circumcenter of $\triangle BHD$, and $O_3$ be the circumcenter of $\triangle HDC$. Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$.

Find all integer pairs $(x,y)$ such that $x^3+y^3=(x+y)^2$.

Determine whether or not there exists a sequence of integers $a_1,a_2,\dots, a_{19}, a_{20}$ such that, the sum of all the terms is negative, and the sum of any three consecutive terms is positive.