MathDB

3

Part of 2010 Slovenia National Olympiad

Problems(4)

Prove that F is the midpoint of AB [Slovenia 2010]

Source:

11/14/2010
Let ABCABC be an isosceles triangle with apex at C.C. Let DD and EE be two points on the sides ACAC and BCBC such that the angle bisectors ∠DEB\angle DEB and ∠ADE\angle ADE meet at F,F, which lies on segment AB.AB. Prove that FF is the midpoint of AB.AB.
trigonometrygeometryangle bisectorgeometry proposed
Prove that AFE and CBD are similar [Slovenia 2010]

Source:

11/14/2010
Let ABCABC be an acute triangle. A line parallel to BCBC intersects the sides ABAB and ACAC at DD and EE, respectively. The circumcircle of the triangle ADEADE intersects the segment CDCD at F (F≠D).F \ (F \neq D). Prove that the triangles AFEAFE and CBDCBD are similar.
geometrycircumcirclegeometry proposed
Points A, E, S and C lie on the same circle [Slovenia 2010]

Source:

11/16/2010
Let ABCABC be an acute triangle with ∣AB∣>∣AC∣.|AB| > |AC|. Let DD be a point on the side ABAB, such that the angles ∠ACD\angle ACD and ∠CBD\angle CBD are equal. Let EE denote the midpoint of BD,BD, and let SS be the circumcenter of the triangle BCD.BCD. Prove that the points A,E,SA, E, S and CC lie on the same circle.
geometrycircumcirclegeometry proposed
Find all functions (y+1)f(x+y) = f(xf(y)) [Slovenia 2010]

Source:

11/16/2010
Find all functions f:[0,+∞)→[0,+∞)f: [0, +\infty) \to [0, +\infty) satisfying the equation (y+1)f(x+y)=f(xf(y))(y+1)f(x+y) = f\left(xf(y)\right) For all non-negative real numbers xx and y.y.
functionalgebra unsolvedalgebra