MathDB

4

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Beautiful geometry

Source: III Caucasus Mathematical Olympiad

3/17/2018
By centroid of a quadrilateral PQRSPQRS we call a common point of two lines through the midpoints of its opposite sides. Suppose that ABCDEFABCDEF is a hexagon inscribed into the circle Ω\Omega centered at OO. Let AB=DEAB=DE, and BC=EFBC=EF. Let XX, YY, and ZZ be centroids of ABDEABDE, BCEFBCEF; and CDFACDFA, respectively. Prove that OO is the orthocenter of triangle XYZXYZ.
geometry
very beatiful problem involving functions

Source: III Caucasus Mathematical Olympiad

3/17/2018
Morteza places a function [0,1]→[0,1][0,1]\to [0,1] (that is a function with domain [0,1] and values from [0,1]) in each cell of an n×nn \times n board. Pavel wants to place a function [0,1]→[0,1][0,1]\to [0,1] to the left of each row and below each column (i.e. to place 2n2n functions in total) so that the following condition holds for any cell in this board: If hh is the function in this cell, ff is the function below its column, and gg is the function to the left of its row, then h(x)=f(g(x))h(x) = f(g(x)) for all x∈[0,1]x \in [0, 1]. Prove that Pavel can always fulfil his plan.
functionalgebra