MathDB

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Part of 2010 Poland - Second Round

Problems(2)

Polish 2nd stage 2010, 1st problem (system of equations)

Source:

11/11/2010
Solve in the real numbers x,y,zx, y, z a system of the equations: {x2−(y+z+yz)x+(y+z)yz=0y2−(z+x+zx)y+(z+x)zx=0z2−(x+y+xy)z+(x+y)xy=0. \begin{cases} x^2 - (y+z+yz)x + (y+z)yz = 0 \\ y^2 - (z + x + zx)y + (z+x)zx = 0 \\ z^2 - (x+y+xy)z + (x+y)xy = 0. \\ \end{cases}
algebra proposedalgebra
Polish 2nd stage 2010, 4th problem (pentagon with one angle)

Source:

11/11/2010
In the convex pentagon ABCDEABCDE all interior angles have the same measure. Prove that the perpendicular bisector of segment EAEA, the perpendicular bisector of segment BCBC and the angle bisector of ∠CDE\angle CDE intersect in one point.
geometryincenterperpendicular bisectorangle bisectorgeometry proposed