3

Part of 1995 Turkey Team Selection Test

Problems(2)

Find the locus of the intersection point

Source: Turkey IMO TST 1995 #3

D

be a point on the small arc

AC

of the circumcircle of an equilateral triangle

ABC

, different from

A

C

E

F

be the projections of

D

BC

AC

respectively. Find the locus of the intersection point of

EF

OD

O

is the center of

ABC

geometrycircumcirclegeometry unsolved

Show there exist a, b such that limit is 1

Source: Turkey IMO TST 1995 #6

\{x_n\}

of real numbers is defined by x_1=1 \text{and} x_{n+1}=x_n+\sqrt[3]{x_n} \text{for} n\geq 1.
Show that there exist real numbers

a, b

\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1

limitalgebra unsolvedalgebra