MathDB

3

Part of 2022 European Mathematical Cup

Problems(2)

Looks like some madness geo again

Source: European Mathematical Cup 2022, Junior Division, Problem 3

12/19/2022
Let ABCABC be an acute-angled triangle with AC>BCAC > BC, with incircle τ\tau centered at II which touches BCBC and ACAC at points DD and EE, respectively. The point MM on τ\tau is such that BM∥DEBM \parallel DE and MM and BB lie on the same halfplane with respect to the angle bisector of ∠ACB\angle ACB. Let FF and HH be the intersections of τ\tau with BMBM and CMCM different from MM, respectively. Let JJ be a point on the line ACAC such that JM∥EHJM \parallel EH. Let KK be the intersection of JFJF and τ\tau different from FF. Prove that ME∥KHME \parallel KH.
geometryincircleParallel Linesangle bisector
Slippery functional equation

Source: European Mathematical Cup 2022, Senior Division, Problem 3

12/20/2022
Determine all functions f:R→Rf: \mathbb{R} \to \mathbb{R} such that f(x3)+f(y)3+f(z)3=3xyz f(x^3) + f(y)^3 + f(z)^3 = 3xyz for all real numbers xx, yy and zz with x+y+z=0x+y+z=0.
functionfunctional equationsubstitutionsCauchy functional equationalgebra