MathDB

7

Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(3)

Another geometry

Source: St Petersburg Olympiad 2014, Grade 11, P7

10/24/2017
II - incenter , MM- midpoint of arc BACBAC of circumcircle, ALAL - angle bisector of triangle ABCABC. MIMI intersect circumcircle in KK. Circumcircle of AKLAKL intersect BCBC at LL and PP. Prove that ∠AIP=90\angle AIP=90
geometryincentercircumcircleangle bisector
Cities and roads

Source: St Petersburg Olympiad 2014, Grade 10, P7

10/26/2017
Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than 11 road.
I think it should be one more condition, like there is cycle that connect all cities
combinatoricsgraph theory
Numbers in the table

Source: St Petersburg Olympiad 2014, Grade 9, P7

10/27/2017
Natural a,b,ca,b,c are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every a×aa \times a, every b×bb \times b, every c×cc \times c squares is even. Is it true, that every number in table must be even?
number theory