MathDB

1

Part of Mathley 2014-15

Problems(4)

collinear wanted, segments with collinear midpoints given, AX//BY// CZ// \ell.

Source: Mathley 2014.1 p1

8/19/2020
Let AD,BE,CFAD, BE, CF be segments whose midpoints are on the same line ℓ\ell. The points X,Y,ZX, Y, Z lie on the lines EF,FD,DEEF, FD, DE respectively such that AX∥BY∥CZ∥ℓAX \parallel BY \parallel CZ \parallel \ell. Prove that X,Y,ZX, Y, Z are collinear.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University
geometrycollinearmidpointparallel
large golden square land lot of 100x100 subdivided into 100 square lots

Source: Mathley 2014.2 p1

8/20/2020
A large golden square land lot of dimension 100×100100 \times 100 m was subdivided into 100100 square lots, each measured 10×1010\times10 m. A king of landfill had his men dump wastes onto some of the lots. There was a practice that if a particular lot was not dumped and twoof its adjacents had waste materials, then the lot would be filled with wastes the next day by the people. One day if all the lotswere filled with wastes, the king would claim his ownership ofthe whole land lot. At least how many lots should have the kind had his men dump wastes onto?
Vu Ha Van, Mathematics Faculty, Yale University, USA.
combinatoricscombinatorial geometry
a copsychus and a sparrow, move across edges of a regular polygon

Source: Mathley 2014.3 p1

8/18/2020
A copsychus and a sparrow, each initially located at one of the vertex of a regular polygon with 103103 edges, fly clockwise to another vertex each. The copsychus moves across ℓ\ell edges each time while the sparrow moves throughd d edges of the polygon, where ℓ≠d\ell \ne d are both integers less than 103103. Assume that, during their journeys, the copsychus has stopped at mm vertices while sparrow has stopped at nn vertices of the polygon, for m≥n≥3m \ge n \ge 3. Determine the value of m,nm, n given that there is only one common single vertex of the polygon that both of birds have stopped at, and there is only one vertex that neither of the birds have reached.
Vu Thi Khoi, Topo University, Hanoi Mathematics Institute, Vietnam, Hoang Qu6c Vietnam, Hanoi.
combinatorics
mathley 2015.3 p1 by Tran Quang Hung

Source:

8/18/2020
Let ABCABC be an acute triangle inscribed in a circle (O)(O) that is fixed, and two of the vertices BB, CC are fixed while vertex AA varies on the circumference of the circle. Let II be the center of the incircle, and ADAD the angle bisector. Let KK, LL be the circumcenters of CADCAD, ABDABD. A line through OO parallel to DLDL, DKDK intersects the line that is through II perpendicular to IBIB, ICIC at MM, NN respectively. Prove that MNMN is tangent to a fixed circle when AA varies on the circle (O)(O).
Tran Quang Hung, Natural Science High School, National University, Hanoi
geometryfixedcircle