MathDB

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Part of 1987 Greece National Olympiad

Problems(3)

max n, convex n-gon with equal diagonals 1987 Greece MO Grade X p1

Source:

9/6/2024
It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural nn such that a convex nn-gon has all it's diagonals equal.
geometrycombinatorial geometryconvex polygon
color plane twith 3 colours 1987 Greece MO Grade XI p1

Source:

9/6/2024
We color all points of a plane using 33 colors. Prove that there are at least two points of the plane having same colours and with distance among them 11.
geometrycombinatoricscombinatorial geometryColoring
gcd = sum =\lambda_i a_i

Source: 1987 Greece MO Grade XII p1

9/6/2024
a) Prove that every sub-group (A,+)(A,+) of group (Z,+)(\mathbb{Z},+) is in the form A=n⋅ZA=n \cdot \mathbb{Z} for some n∈Zn \in \mathbb{Z} where n⋅Z={n⋅x/x∈Z}n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}.
b) Using problem (a) , prove that the greatest common divisor dd of non zero integers a1,a2,...,ana_1, a_2,... ,a_n is given by relation d=λ1a1+λ2a2+...λnand=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n with λi∈Z,  i=1,2,...,n\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n
group theoryabstract algebragreatest common divisornumber theory