MathDB

3

Part of 1993 Romania Team Selection Test

Problems(4)

partition of 1,2,...,2^n into 2 classes, none has arithmetic progression

Source: Romania BMO TST 1993 p3

2/17/2020
Show that the set {1,2,....,2n}\{1,2,....,2^n\} can be partitioned in two classes, none of which contains an arithmetic progression of length 2n2n.
arithmetic sequencepartitionSubsetscombinatorics
does the hexagon necessarily have a center of symmetry ?

Source: Romania IMO TST 1993 1.3

2/17/2020
Suppose that each of the diagonals AD,BE,CFAD,BE,CF divides the hexagon ABCDEFABCDEF into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?
symmetryhexagongeometrydiagonalsequal areas
P={1,2,...,p-1}

Source: Romania TST 1993

8/4/2009
Let p≥5 p\geq 5 be a prime number.Prove that for any partition of the set P\equal{}\{1,2,3,...,p\minus{}1\} in 3 3 subsets there exists numbers x,y,z x,y,z each belonging to a distinct subset,such that x\plus{}y\equiv z (mod p)
searchnumber theory unsolvednumber theory
any A in B, 3 points X,Y,Z \in B with AX = AY = AZ = 1, B set of n points

Source: Romania IMO TST 1993 2.3

2/17/2020
Find all integers n>1n > 1 for which there is a set BB of nn points in the plane such that for any A∈BA \in B there are three points X,Y,Z∈BX,Y,Z \in B with AX=AY=AZ=1AX = AY = AZ = 1.
combinatoricscombinatorial geometry