MathDB

2

Part of 2015 Belarus Team Selection Test

Problems(4)

numbers 1,..,9 in a 3x3 table, increase or decrease by one 1 all numbers in 2x2

Source: 2015 Belarus TST 1.2

11/7/2020
All the numbers 1,2,...,91,2,...,9 are written in the cells of a 3×33\times 3 table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary 2×22\times2 square of the table and either decrease by 11 or increase by 11 all four numbers of the square. After some number of such moves all numbers of the table become equal to some number aa. Find all possible values of aa.
I.Voronovich
combinatorics
numbers 2,0,1,5 occur in sequence of last digits 2,0,2,9,3,...

Source: 2015 Belarus TST 2.2

11/5/2020
In the sequence of digits 2,0,2,9,3,...2,0,2,9,3,... any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers 2,0,1,52,0,1,5 in that order occur in the sequence?
Folklore
Sequencenumber theoryLast digitDigits
one circle with diameter median tangent to altitude, the other one also

Source: Belarus TST 2015 3.2

6/9/2020
The medians AMAM and BNBN of a triangle ABCABC are the diameters of the circles ω1\omega_1 and ω2\omega_2. If ω1\omega_1 touches the altitude CHCH, prove that ω2\omega_2 also touches CHCH.
I. Gorodnin
geometrytangentcirclesMedians
circumcenter of triangle AMN belong to segment AC, DM+BN=MN, AB=AD

Source: Belarus TST 2015 7.2

6/9/2020
Given a cyclic ABCDABCD with AB=ADAB=AD. Points MM and NN are marked on the sides CDCD and BCBC, respectively, so that DM+BN=MNDM+BN=MN. Prove that the circumcenter of the triangle AMNAMN belongs to the segment ACAC.
N.Sedrakian
geometryCircumcentercyclic quadrilateralequal segments