Bosnia Herzegovina Team Selection Test 2014

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 $OI = IH$.

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Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,1,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,40,Italy,1,Itan,1,Japan,1,JBMO,2,Kazakhstan,1,Korea,2,Macedonia,1,Mediterrane,7,Mediterranean,3,MEMO,1,Mexico,1,Miklós Schweitzer,1,Moldova,1,National,29,Olympic Revenge,1,Paenza,1,Paraguayan,1,Rusia,1,TST,9,Turkey,1,Tuymaada,19,Undergraduate,1,Zhautykov,1,
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MATHEMATICAL OLYMPIAD PROBLEMS: Bosnia Herzegovina Team Selection Test 2014
Bosnia Herzegovina Team Selection Test 2014
MATHEMATICAL OLYMPIAD PROBLEMS
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