MathDB

6

Part of 2008 All-Russian Olympiad

Problems(3)

Parallel

Source: All Russian Mathematical Olympiad 2008. 10.6

6/13/2008
In a scalene triangle ABC ABC the altitudes AA1 AA_{1} and CC1 CC_{1} intersect at H,O H, O is the circumcenter, and B0 B_{0} the midpoint of side AC AC. The line BO BO intersects side AC AC at P P, while the lines BH BH and A1C1 A_{1}C_{1} meet at Q Q. Prove that the lines HB0 HB_{0} and PQ PQ are parallel.
geometrycircumcirclegeometric transformationreflectiontrigonometrygeometry proposed
9th grade 2008 Russian Mathematical Olympiad 6th question

Source: All Russian 2008, Grade 9, Problem 6

5/21/2008
The incircle of a triangle ABCABC touches the side ABAB and ACAC at respectively at XX and YY. Let KK be the midpoint of the arc AB^\widehat{AB} on the circumcircle of ABCABC. Assume that XYXY bisects the segment AKAK. What are the possible measures of angle BACBAC?
geometrycircumcircletrigonometrysearchgeometry unsolved
Magician

Source: All Russian Mathematical Olympiad 2008. 11.6

6/13/2008
A magician should determine the area of a hidden convex 2008 2008-gon A1A2⋯A2008 A_{1}A_{2}\cdots A_{2008}. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in 2006 2006 steps.
geometryperimeteralgorithmcombinatorics proposedcombinatorics