MathDB

2

Part of 2004 All-Russian Olympiad

Problems(5)

tangent quadrilateral

Source: Russian Olympiad 2004, problem 9.2

5/3/2004
Let ABCDABCD be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles DABDAB and ABCABC intersect each other at KK; the exterior angle bisectors of the angles ABCABC and BCDBCD intersect each other at LL; the exterior angle bisectors of the angles BCDBCD and CDACDA intersect each other at MM; the exterior angle bisectors of the angles CDACDA and DABDAB intersect each other at NN. Let K1K_{1}, L1L_{1}, M1M_{1} and N1N_{1} be the orthocenters of the triangles ABKABK, BCLBCL, CDMCDM and DANDAN, respectively. Show that the quadrilateral K1L1M1N1K_{1}L_{1}M_{1}N_{1} is a parallelogram.
geometryincenterexterior anglecomplex numbersgeometry solved
in the cabinet 2004 telephones are located

Source: Russian Olympiad 2004, problem 9.6

5/3/2004
In the cabinet 2004 telephones are located; each two of these telephones are connected by a cable, which is colored in one of four colors. From each color there is one cable at least. Can one always select several telephones in such a way that among their pairwise cable connections exactly 3 different colors occur?
inductioncombinatorics unsolvedcombinatorics
the centers of the excircles of a triangle ABC

Source: Russian Olympiad 2004, problem 11.2

5/4/2004
Let I(A) I(A) and I(B) I(B) be the centers of the excircles of a triangle ABC, ABC, which touches the sides BC BC and CA CA in its interior. Furthermore let P P a point on the circumcircle ω \omega of the triangle ABC. ABC. Show that the center of the segment which connects the circumcenters of the triangles I(A)CP I(A)CP and I(B)CP I(B)CP coincides with the center of the circle ω. \omega.
geometrycircumcircleEulertrapezoidincenterrectanglegeometric transformation
a country has 1001 cities, and each two cities are connected

Source: Russian Olympiad 2004, problem 10.6

5/3/2004
A country has 1001 cities, and each two cities are connected by a one-way street. From each city exactly 500 roads begin, and in each city 500 roads end. Now an independent republic splits itself off the country, which contains 668 of the 1001 cities. Prove that one can reach every other city of the republic from each city of this republic without being forced to leave the republic.
graph theorycombinatorics unsolvedcombinatorics
pairwise non-collinear vectors of the plane

Source: Russian Olympiad 2004, problem 11.6

5/4/2004
Prove that there is no finite set which contains more than 2N, 2N, with N>3, N > 3, pairwise non-collinear vectors of the plane, and to which the following two characteristics apply: 1) for N N arbitrary vectors from this set there are always further N\minus{}1 vectors from this set so that the sum of these is 2N\minus{}1 vectors is equal to the zero-vector; 2) for N N arbitrary vectors from this set there are always further N N vectors from this set so that the sum of these is 2N 2N vectors is equal to the zero-vector.
vectorgeometry unsolvedgeometry