MathDB

2

Part of 2011 Tuymaada Olympiad

Problems(3)

Making an 11x11 hole in a 2011x2011 board tiled with dominos

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

7/29/2011
How many ways are there to remove an 11×1111\times11 square from a 2011×20112011\times2011 square so that the remaining part can be tiled with dominoes (1×21\times 2 rectangles)?
geometryrectanglecombinatorics
Parallelism in a cyclic quad, circle & diagonals diagram

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

7/29/2011
A circle passing through the vertices AA and BB of a cyclic quadrilateral ABCDABCD intersects diagonals ACAC and BDBD at EE and FF, respectively. The lines AFAF and BCBC meet at a point PP, and the lines BEBE and ADAD meet at a point QQ. Prove that PQPQ is parallel to CDCD.
geometrycircumcirclecyclic quadrilateralprojective geometrygeometry unsolved
Lines thru the midpoint of the common chord of two circles

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

7/29/2011
Circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB, and MM is the midpoint of ABAB. Points S1S_1 and S2S_2 lie on the line ABAB (but not between AA and BB). The tangents drawn from S1S_1 to ω1\omega_1 touch it at X1X_1 and Y1Y_1, and the tangents drawn from S2S_2 to ω2\omega_2 touch it at X2X_2 and Y2Y_2. Prove that if the line X1X2X_1X_2 passes through MM, then line Y1Y2Y_1Y_2 also passes through MM.
geometrygeometric transformationreflectiongeometry unsolved