MathDB

6

Part of 2010 Tournament Of Towns

Problems(4)

Finding least number of lines formed by 31 segments...

Source:

2/8/2011
A broken line consists of 3131 segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
geometry unsolvedgeometry
Tournament of the Towns SP2010 (I guess), Trapezoid

Source:

5/8/2010
Quadrilateral ABCDABCD is circumscribed around the circle with centre II. Let points MM and NN be the midpoints of sides ABAB and CDCD respectively and let IMAB=INCD\frac{IM}{AB} = \frac{IN}{CD}. Prove that ABCDABCD is either a trapezoid or a parallelogram.
geometrytrapezoidratiogeometric transformationreflectionincenteranalytic geometry
KB=KC in triangle ABC, K is intersection of perpendiculars.

Source:

2/13/2011
In acute triangle ABCABC, an arbitrary point PP is chosen on altitude AHAH. Points EE and FF are the midpoints of sides CACA and ABAB respectively. The perpendiculars from EE to CPCP and from FF to BPBP meet at point KK. Prove that KB=KCKB = KC.
geometrygeometric transformationreflectionincenterperpendicular bisectorgeometry unsolved
Removing 990 rows or columns to get at least one 1 or 0...

Source:

2/19/2011
Each cell of a 1000×10001000\times 1000 table contains 00 or 11. Prove that one can either cut out 990990 rows so that at least one 11 remains in each column, or cut out 990990 columns so that at least one 00 remains in each row.
geometryrectanglecombinatorics unsolvedcombinatorics