MathDB

3

Part of 2020 Tournament Of Towns

Problems(4)

orthocenter wanted, parallelogram and rhombus related

Source: Tournament of Towns, Junior O-Level Paper, Spring 2020 , p3

6/3/2020
Let ABCDABCD be a rhombus, let APQCAPQC be a parallelogram such that the point BB lies inside it and the side APAP is equal to the side of the rhombus. Prove that BB is the orthocenter of the triangle DPQDPQ.
Egor Bakaev
geometryrhombusparallelogramorthocenter
inscribed N-gon with different sides and integer angles (N=19,20)

Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p3

6/10/2020
Is it possible to inscribe an NN-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) N=19N = 19, b) N=20N = 20 ?
Mikhail Malkin
CyclicinscribedpolygonIntegergeometrycombinatorial geometrycombinatorics
41 letters on a circle, each letter is A or B

Source: Tournament of Towns, Senior Ο-Level Paper, Spring 2020 , p3

6/4/2020
There are 4141 letters on a circle, each letter is AA or BB. It is allowed to replace ABAABA by BB and conversely, as well as to replace BABBAB by AA and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations?
Maxim Didin
combinatorics
rectangle that contains 20 marked cells out of 40 cells on infinite chessboard

Source: Tournament of Towns 2020 oral p3 (15 March 2020)

5/18/2020
4040 cells were marked on an infinite chessboard. Is it always possible to find a rectangle that contains 2020 marked cells?
M. Evdokimov
combinatoricscombinatorial geometrygrid