MathDB

3

Part of 2019 Brazil National Olympiad

Problems(2)

Hard functional equation

Source: Brazil National Olympiad 2019 #3

11/14/2019
Let R>0\mathbb{R}_{>0} be the set of the positive real numbers. Find all functions f:R>0→R>0f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} such that f(xy+f(x))=f(f(x)f(y))+xf(xy+f(x))=f(f(x)f(y))+x for all positive real numbers xx and yy.
functional equationalgebra
Lines contained in one point

Source: Brazil National Olympiad 2019 - level 2 - #3

11/23/2019
Let ABCABC be an acutangle triangle inscribed in a circle Γ\Gamma of center OO. Let DD be the height of the vertex AA. Let E and F be points over Γ\Gamma such that AE=AD=AFAE = AD = AF. Let PP and QQ be the intersection points of the EFEF with sides ABAB and ACAC respectively. Let XX be the second intersection point of Γ\Gamma with the circle circumscribed to the triangle APQAP Q. Show that the lines XDXD and AOAO meet at a point above sobre
euclidean geometrygeometryBrazilian Math Olympiad