MathDB

2

Part of 2020 Taiwan TST Round 1

Problems(2)

Bisecting Segment

Source: 2020 Taiwan TST Round 1 Independent Study 2-2

5/23/2020
Let point HH be the orthocenter of a scalene triangle ABCABC. Line AHAH intersects with the circumcircle Ω\Omega of triangle ABCABC again at point PP. Line BH,CHBH, CH meets with AC,ABAC,AB at point EE and FF, respectively. Let PE,PFPE, PF meet Ω\Omega again at point Q,RQ,R, respectively. Point YY lies on Ω\Omega so that lines AY,QRAY,QR and EFEF are concurrent. Prove that PYPY bisects EFEF.
geometrycircumcircle
A simple functional equation

Source: 2020 Taiwan TST Round 1 Mock Exam 2

5/23/2020
Let R\mathbb{R} be the set of all real numbers. Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that for any x,y∈Rx,y\in \mathbb{R}, there holds f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).
functionalgebrafunctional equationTaiwan