MathDB

3

Part of 2001 All-Russian Olympiad

Problems(5)

K inside parallelogram ABCD => NAK = NCK

Source: All-Russian MO 2001 Grade 9 #3

1/2/2012
A point KK is taken inside parallelogram ABCDABCD so that the midpoint of ADAD is equidistant from KK and CC, and the midpoint of CDCD is equidistant form KK and AA. Let NN be the midpoint of BKBK. Prove that the angles NAKNAK and NCKNCK are equal.
geometryparallelogramcircumcirclegeometry unsolved
Russian geometry

Source:

7/27/2003
Let NN be a point on the longest side ACAC of a triangle ABCABC. The perpendicular bisectors of ANAN and NCNC intersect ABAB and BCBC respectively in KK and MM. Prove that the circumcenter OO of △ABC\triangle ABC lies on the circumcircle of triangle KBMKBM.
geometrycircumcircleparallelogramsimilar triangles
orthocenter

Source:

12/17/2011
Points A1,B1,C1A_1, B_1, C_1 inside an acute-angled triangle ABCABC are selected on the altitudes from A,B,CA, B, C respectively so that the sum of the areas of triangles ABC1,BCA1ABC_1, BCA_1, and CAB1CAB_1 is equal to the area of triangle ABCABC. Prove that the circumcircle of triangle A1B1C1A_1B_1C_1 passes through the orthocenter HH of triangle ABCABC.
geometrycircumcirclefunctioncyclic quadrilateralgeometry unsolved
Convex polygons

Source: Russian 2001

4/5/2006
There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.
combinatorics unsolvedcombinatorics
2001 towns in a country

Source: All-Russian MO 2001 Grade 11 #7

1/3/2012
The 20012001 towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set DD of towns dominant if every town not in DD is connected by a road to a town in DD. Suppose that each dominant set consists of at least kk towns. Prove that the country can be partitioned into 2001−k2001-k republics in such a way that no two towns in the same republic are connected by a road.
graph theorycombinatorics unsolvedcombinatorics