MathDB

2

Part of 1997 Polish MO Finals

Problems(2)

system of equations

Source:

11/11/2005
Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}
inequalitiesalgebra unsolvedalgebra
Convex pentagon - equality of angles

Source:

11/11/2005
ABCDEABCDE is a convex pentagon such that DC=DEDC = DE and ∠C=∠E=90⋅\angle C = \angle E = 90^{\cdot}. FF is a point on the side ABAB such that AFBF=AEBC\frac{AF}{BF}= \frac{AE}{BC}. Show that ∠FCE=∠ADE\angle FCE = \angle ADE and ∠FEC=∠BDC\angle FEC = \angle BDC.
geometry solvedgeometry