MathDB

3

Part of 2010 Polish MO Finals

Problems(2)

Polish MO Final 2010, 3rd problem (parallelogram and circle)

Source:

11/7/2010
ABCDABCD is a parallelogram in which angle DABDAB is acute. Points A,P,B,DA, P, B, D lie on one circle in exactly this order. Lines APAP and CDCD intersect in QQ. Point OO is the circumcenter of the triangle CPQCPQ. Prove that if D≠OD \neq O then the lines ADAD and DODO are perpendicular.
geometryparallelogramcircumcircletrigonometrytrapezoidtrig identitiesLaw of Sines
Polish MO Final 2010, 6th problem (sequence with properties)

Source:

11/7/2010
Real number C>1C > 1 is given. Sequence of positive real numbers a1,a2,a3,…a_1, a_2, a_3, \ldots, in which a1=1a_1=1 and a2=2a_2=2, satisfy the conditions amn=aman,a_{mn}=a_ma_n, am+n≤C(am+an),a_{m+n} \leq C(a_m + a_n), for m,n=1,2,3,…m, n = 1, 2, 3, \ldots. Prove that an=na_n = n for n=1,2,3,…n=1, 2, 3, \ldots.
logarithmslimitalgebra proposedalgebraInequalityinequalities