MathDB

2

Part of 2019 Philippine TST

Problems(2)

11 labelled points and a harmonic length condition

Source: 2019 Philippine IMO TST1 Problem 2

5/4/2022
In a triangle ABCABC with circumcircle Γ\Gamma, MM is the midpoint of BCBC and point DD lies on segment MCMC. Point GG lies on ray BC→\overrightarrow{BC} past CC such that BCDC=BGGC\frac{BC}{DC} = \frac{BG}{GC}, and let NN be the midpoint of DGDG. The points PP, SS, and TT are defined as follows: [list = i] [*] Line CACA meets the circumcircle Γ1\Gamma_1 of triangle AGDAGD again at point PP. [*] Line PMPM meets Γ1\Gamma_1 again at SS. [*] Line PNPN meets the line through AA that is parallel to BCBC at QQ. Line CQCQ meets Γ\Gamma again at TT.
Prove that the points PP, SS, TT, and CC are concyclic.
geometryharmonic bundlescircumcirclegeometry unsolved
Degree 2019 functional equation

Source: 2019 Philippine IMO TST3 Problem 2

5/4/2022
Find all functions f:R→Rf : \mathbb{R} \to \mathbb{R} that satisfy the equation f(x2019+y2019)=x(f(x))2018+y(f(y))2018f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018} for all real numbers xx and yy.
algebrafunctional equation