MathDB

2

Part of 2010 Poland - Second Round

Problems(2)

Polish 2nd stage 2010, 2nd problem (tetrahedron)

Source:

11/11/2010
The orthogonal projections of the vertices A,B,CA, B, C of the tetrahedron ABCDABCD on the opposite faces are denoted by A′,B′,C′A', B', C' respectively. Suppose that point A′A' is the circumcenter of the triangle BCDBCD, point B′B' is the incenter of the triangle ACDACD and C′C' is the centroid of the triangle ABDABD. Prove that tetrahedron ABCDABCD is regular.
geometry3D geometrytetrahedroncircumcircleincentergeometry proposed
Polish 2nd stage 2010, 5th problem (functional equation)

Source:

11/11/2010
Find all monotonic functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} satisfying f(f(x)−y)+f(x+y)=0,f(f(x) - y) + f(x+y) = 0, for every real x,yx, y. (Note that monotonic means that function is not increasing or not decreasing)
functionalgebra proposedalgebra