MathDB

1

Part of 1989 IberoAmerican

Problems(2)

x,y,z system of equations

Source: IberoAmerican 1989 Q1

11/27/2010
Determine all triples of real numbers that satisfy the following system of equations: x+y−z=−1x2−y2+z2=1−x3+y3+z3=−1x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1
algebrasystem of equationsalgebra proposed
Prove M P · OA = BC · OQ

Source: IberoAmerican 1989 Q4

11/27/2010
The incircle of the triangle ABCABC is tangent to sides ACAC and BCBC at MM and NN, respectively. The bisectors of the angles at AA and BB intersect MNMN at points PP and QQ, respectively. Let OO be the incentre of △ABC\triangle ABC. Prove that MP⋅OA=BC⋅OQMP\cdot OA=BC\cdot OQ.
geometryincentergeometry proposed