MathDB

1

Part of 2014 Bosnia Herzegovina Team Selection Test

Problems(2)

Easy circle and tangents

Source: Bosnia and Herzegovina TST 2014 day 1 problem 1

5/10/2014
Let kk be the circle and AA and BB points on circle which are not diametrically opposite. On minor arc ABAB lies point arbitrary point CC. Let DD, EE and FF be foots of perpendiculars from CC on chord ABAB and tangents of circle kk in points AA and BB. Prove that CD=CE⋅CFCD= \sqrt {CE \cdot CF}
trigonometrygeometry unsolvedgeometry
Easy sequence

Source: Bosnia and Herzegovina TST 2014 day 2 problem 1

5/11/2014
Sequence ana_n is defined by a1=12a_1=\frac{1}{2}, am=am−12m⋅am−1+1a_m=\frac{a_{m-1}}{2m \cdot a_{m-1} + 1} for m>1m>1. Determine value of a1+a2+...+aka_1+a_2+...+a_k in terms of kk, where kk is positive integer.
algebra solvedalgebra