MathDB

6

Part of 2019 Saint Petersburg Mathematical Olympiad

Problems(3)

2019 Saint Petersburg Grade 11 P6

Source: Saint Petersburg 2019

4/14/2019
Supppose that there are roads ABAB and CDCD but there are no roads BCBC and ADAD between four cities AA, BB, CC, and DD. Define restructing to be the changing a pair of roads ABAB and CDCD to the pair of roads BCBC and ADAD. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly 100100 roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly 100100 roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings.
(Т. Зубов)
Thanks to the user Vlados021 for translating the problem.
combinatorics
arranging all positive integers in an infinite chessboard

Source: St. Petersburg 2019 10.6

5/1/2019
Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each nn in each square n×nn \times n the sum of the numbers is a multiple of nn?
combinatorial geometrycombinatoricspositive integersChessboardinfinite chessboard
line passes through the intersection of two circumcircles, when <A=60^O

Source: StT. Petersburg 2019 9.6

5/1/2019
The bisectors BB1BB_1 and CC1CC_1 of the acute triangle ABCABC intersect in point II. On the extensions of the segments BB1BB_1 and CC1CC_1, the points B′B' and C′C' are marked, respectively So, the quadrilateral AB′IC′AB'IC' is a parallelogram. Prove that if ∠BAC=60o\angle BAC = 60^o, then the straight line B′C′B'C' passes through the intersection point of the circumscribed circles of the triangles BC1B′BC_1B' and CB1C′CB_1C'.
geometryparallelogramangle bisectorcircumcircle