MathDB

1

Part of 2011 Poland - Second Round

Problems(2)

System of equations

Source: Polish MO second round 2011

2/19/2012
For x,y∈Rx,y\in\mathbb{R}, solve the system of equations {(x−y)(x3+y3)=7(x+y)(x3−y3)=3 \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases}
algebra unsolvedalgebra
Points on semicircle

Source: Polish MO second round 2011

2/19/2012
Points A,B,C,D,E,FA,B,C,D,E,F lie in that order on semicircle centered at OO, we assume that AD=BE=CFAD=BE=CF. GG is a common point of BEBE and ADAD, HH is a common point of BEBE and CDCD. Prove that: ∠AOC=2∠GOH.\angle AOC=2\angle GOH.
geometrygeometric transformationrotationgeometry unsolved