Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $... 1. Let$k$be the circle and$A$and$B$points on circle which are not diametrically opposite. On minor arc$AB$lies point arbitrary point$C$. Let$D$,$E$and$F$be foots of perpendiculars from$C$on chord$AB$and tangents of circle$k$in points$A$and$B$. Prove that$CD= \sqrt {CE \cdot CF}$2. Let$a$,$b$and$c$be distinct real numbers. a) Determine value of $$\frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b}.$$ b) Determine value of $$\frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b}.$$ c) Prove the following inequality $$\frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2}.$$ When does equality holds? 3. Find all nonnegative integer numbers such that $$7^x- 2 \cdot 5^y = -1$$ 4. Sequence$a_n$is defined by $$a_1=\frac{1}{2},\quad a_m=\frac{a_{m-1}}{2m \cdot a_{m-1} + 1},\,m>1.$$ Determine value of$a_1+a_2+...+a_k$in terms of$k$, where$k$is positive integer. 5. It is given regular$n$-sided polygon,$n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon? 6. Let$D$and$E$be foots of altitudes from$A$and$B$of triangle$ABC$,$F$be intersection point of angle bisector from$C$with side$AB$, and$O$,$I$and$H$be circumcenter, center of inscribed circle and orthocenter of triangle$ABC$, respectively. If $$\frac{CF}{AD}+ \frac{CF}{BE}=2,$$ prove that$OI = IH\$.

Name