MathDB

3

Part of 2012 Middle European Mathematical Olympiad

Problems(2)

Middle European Mathematical Olympiad 2012 - Individuals I-3

Source: Middle European Mathematical Olympiad 2012 - Individuals I-3

9/14/2012
In a given trapezium ABCD ABCD with AB AB parallel to CD CD and AB>CD AB > CD , the line BD BD bisects the angle ∠ADC \angle ADC . The line through C C parallel to AD AD meets the segments BD BD and AB AB in E E and F F , respectively. Let O O be the circumcenter of the triangle BEF BEF . Suppose that ∠ACO=60∘ \angle ACO = 60^{\circ} . Prove the equality CF=AF+FO. CF = AF + FO .
geometrytrapezoidcircumcirclegeometry proposed
Considering the number of words with parity

Source: Middle European Mathematical Olympiad 2012 - Team Compt. T-3

9/14/2012
Let n n be a positive integer. Consider words of length nn composed of letters from the set {M,E,O} \{ M, E, O \} . Let a a be the number of such words containing an even number (possibly 0) of blocks ME ME and an even number (possibly 0) blocks of MO MO . Similarly let b b the number of such words containing an odd number of blocks ME ME and an odd number of blocks MO MO . Prove that a>b a>b .
combinatorics proposedcombinatorics