The sides of a triangle $arcx,2x+1$ and $x+2$ for some positive rational number $x.$ If one angle of the triangle is $60°,$ what is the perimeter of the triangle?

Let $ABC$ be a triangle with $\angle BAC=90°$ and $D$ be the point on the side $BC$ such that $AD\perp BC.$ Let $r, r_1$ and $r_2$ be the inradius of triangles $ABC, ABD,$ and $ACD$, respectively. If $r, r_1,$ and $r_2$ are positive integers and one of them is $5$ find the largest possible value of $r+r_1+r_2$.

A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid-point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass (See the figure.). If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d?$ (ignore the thickness of the glass.)