MathDB

3

Part of 2011 Tuymaada Olympiad

Problems(4)

Midpoints of chords of an excircle through the circumcircle

Source: XVIII Tuymaada Mathematical Olympiad (2011), Junior Level

7/29/2011
An excircle of triangle ABCABC touches the side ABAB at PP and the extensions of sides ACAC and BCBC at QQ and RR, respectively. Prove that if the midpoint of PQPQ lies on the circumcircle of ABCABC, then the midpoint of PRPR also lies on that circumcircle.
geometrycircumcirclegeometry unsolved
100 & 101 moves for a knight to reach squares in chessboard

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

7/29/2011
Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square OO. A square is called singular if 100100 is written in it and 101101 is written in all four squares sharing a side with it. How many singular squares are there?
combinatorics unsolvedcombinatorics
Switching letters so that long word is not periodic

Source: XVIII Tuymaada Mathematical Olympiad (2011)

7/29/2011
In a word of more than 1010 letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not periodic, that is, cannot be divided into equal subwords.
combinatorics unsolvedcombinatorics
Collinear (diagonal intersect perp bisector)'s in a heaxgon

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

7/29/2011
In a convex hexagon AC′BA′CB′AC'BA'CB', every two opposite sides are equal. Let A1A_1 denote the point of intersection of BCBC with the perpendicular bisector of AA′AA'. Define B1B_1 and C1C_1 similarly. Prove that A1A_1, B1B_1, and C1C_1 are collinear.
geometryperpendicular bisectorgeometry unsolved