Embibe Experts Solutions for Chapter: Polygon, Exercise 1: IOQM - PRMO and RMO 2019 (25-08-2019)

Let $ABCD$ be a square with side length $100.$  A circle with centre $C$ and radius $CD$ is drawn. Another circle of radius $r,$ lying inside $ABCD,$ is drawn to touch this circle externally and such that the circle also touches $AB$ and $AD.$ If $r=m+n\sqrt{k}.$ Where $m, n$ are integers and $k$ is a prime number, find the value of $\frac{m+n}{k}.$

Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28°.$ What is the smallest possible value of $n$?

From a square with sides of length $5,$ triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer?

Let $t$ be the area of a regular pentagon with each side equal to $1.$ Let $P\left(x\right)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $\text{gcd}$ of all the coefficients equal to $1 .$ If $M$ is the sum of the absolute values of the coefficients of $P\left(x\right).$ What is the integer closest to $\sqrt{M}? \left[\sin 18°=\left(\frac{\sqrt{5}-1}{4}\right)\right]$

A point $P$ in the interior of a regular hexagon is at distance $8,8,16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is radius of the circumscribed circle of the hexagon, what is the integer closest to $r?$