MathDB

2

Part of 2002 All-Russian Olympiad

Problems(5)

O1A = O2A => ABC Isosceles

Source: 2002 All-Russian MO, Grade 9, Problem 2

1/2/2012
Point AA lies on one ray and points B,CB,C lie on the other ray of an angle with the vertex at OO such that BB lies between OO and CC. Let O1O_1 be the incenter of △OAB\triangle OAB and O2O_2 be the center of the excircle of △OAC\triangle OAC touching side ACAC. Prove that if O1A=O2AO_1A = O_2A, then the triangle ABCABC is isosceles.
geometryincenterangle bisectorgeometry unsolved
One red cell, k blue cells, and a pack of 2n cards

Source: 2002 All-Russian MO, Grade 9, Problem 6; Grade 10, Problem 6

1/2/2012
We are given one red and k>1k>1 blue cells, and a pack of 2n2n cards, enumerated by the numbers from 11 to 2n2n. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by 11, or onto an empty cell. Given kk, what is the maximal nn for which it is always possible to move all the cards onto a blue cell?
inductionpigeonhole principlecombinatorics unsolvedcombinatorics
orthogonal coordinate from Russia

Source: Challenging.

4/9/2008
Several points are given in the plane. Suppose that for any three of them, there exists an orthogonal coordinate system (determined by the two axes and the unit length) in which these three points have integer coordinates. Prove that there exists an orthogonal coordinate system in which all the given points have integer coordinates.
analytic geometrycombinatorics unsolvedcombinatorics
Show cyclic ABCD is trapezoid

Source: All-Russian MO 2002 Grade 10 #2

1/2/2012
A quadrilateral ABCDABCD is inscribed in a circle ω\omega. The tangent to ω\omega at AA intersects the ray CBCB at KK, and the tangent to ω\omega at BB intersects the ray DADA at MM. Prove that if AM=ADAM=AD and BK=BCBK=BC, then ABCDABCD is a trapezoid.
geometrytrapezoidtrigonometryparallelogramgeometric transformationreflectiontrig identities
Need euclidean prove

Source:

1/1/2008
The diagonals ACAC and BDBD of a cyclic quadrilateral ABCDABCD meet at OO. The circumcircles of triangles AOBAOB and CODCOD intersect again at KK. Point LL is such that the triangles BLCBLC and AKDAKD are similar and equally oriented. Prove that if the quadrilateral BLCKBLCK is convex, then it is tangent [has an incircle].
geometrycircumcirclegeometric transformationcyclic quadrilateralgeometry unsolved