MathDB

2

Part of 1987 IberoAmerican

Problems(2)

AMPN is inscriptable implies ABC is isosceles

Source: IberoAmerican 1987 Q2

11/12/2010
In a triangle ABCABC, MM and NN are the respective midpoints of the sides ACAC and ABAB, and PP is the point of intersection of BMBM and CNCN. Prove that, if it is possible to inscribe a circle in the quadrilateral AMPNAMPN, then the triangle ABCABC is isosceles.
geometrygeometric transformationreflectionconicshyperbolasymmetrytrigonometry
Find arctan r + arctan s +arctan t

Source: IberoAmerican 1987 Q5

11/12/2010
Let r,s,tr,s,t be the roots of the equation x(x−2)(3x−7)=2x(x-2)(3x-7)=2. Show that r,s,tr,s,t are real and positive and determine arctan⁡r+arctan⁡s+arctan⁡t\arctan r+\arctan s +\arctan t.
algebra unsolvedalgebra