MathDB

1

Part of 2007 JBMO Shortlist

Problems(3)

İnterior point

Source: JBMO Shortlist 2007

5/12/2016
G1\boxed{\text{G1}} Let MM be interior point of the triangle ABCABC with AB=MCAB=MC
geometry
tilings with m x n rectangles with corners

Source: JBMO Shortlist 2007 C1

10/14/2017
We call a tiling of an m×nm \times n rectangle with corners (see figure below) "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some mm and nn there exists a "regular" tiling of the m×nm \times n rectangular then there exists a "regular" tiling also for the 2m×2n2m \times 2n rectangle.
JBMOcombinatorics
1/x+1/y+1/[x, y]+1/(x, y)=1/2

Source: JBMO Shortlist 2007 N1

10/14/2017
Find all the pairs positive integers (x,y)(x, y) such that 1x+1y+1[x,y]+1(x,y)=12\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2} , where (x,y)(x, y) is the greatest common divisor of x,yx, y and [x,y][x, y] is the least common multiple of x,yx, y.
least common multiplegreatest common divisorJBMOnumber theory