MathDB

1

Part of 2019 All-Russian Olympiad

Problems(2)

ARMO 2019 9.1

Source: All-Russian Math Olympiad 2019

7/11/2019
There are 5 points on plane. Prove that you can chose some of them and shift them such that distances between shifted points won't change and as a result there will be symetric by some line set of 5 points.
ARMOcombinatoricscombinatorial geometry
ARMO 2019 11.1

Source: All-Russian Math Olimpiad 2019, http://vos.olimpiada.ru/main/table/tasks/#table

7/12/2019
There is located real number f(A)f(A) in any point A on the plane. It's known that if MM will be centroid of triangle ABCABC then f(M)=f(A)+f(B)+f(C)f(M)=f(A)+f(B)+f(C). Prove that f(A)=0f(A)=0 for all points A.
ARMO