MathDB

23

Part of 2019 India PRMO

Problems(2)

Geometry / Cyclic Quadilateral

Source:

8/11/2019
Let ABCDABCD be a convex cyclic quadilateral. Suppose PP is a point in the plane of the quadilateral such that the sum of its distances from the vertices of ABCDABCD is the least. If {PC,PB,PC,PD}={3,4,6,8}\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}, what is the maxumum possible area of ABCDABCD?
geometrycyclic quadrilateral
Polynomial With Trigo Roots

Source: PRMO Leg 2 P23

8/25/2019
Let tt be the area of a regular pentagon with each side equal to 11. Let P(x)=0P(x)=0 be the polynomial equation with least degree, having integer coefficients, satisfied by x=tx=t and the gcd⁡\gcd of all the coefficients equal to 11. If MM is the sum of the absolute values of the coefficients of P(x)P(x), What is the integer closest to M\sqrt{M} ? (sin⁡18∘=(5−1)/2\sin 18^{\circ}=(\sqrt{5}-1)/2)
algebrapolynomial