Day 1 - 10 May 2014
1 Let 
be the circle and 
and 
points on circle which are not diametrically opposite. On minor arc 
lies point arbitrary point 
. Let 
, 
and 
be foots of perpendiculars from on chord and tangents of circle in points and . Prove that 
.
be the circle and and points on circle which are not diametrically opposite. On minor arc lies point arbitrary point . Let , and be foots of perpendiculars from on chord and tangents of circle in points and . Prove that .
Determine value of 
.
Prove the following ineqaulity
.
When does eqaulity holds?
3 Find all nonnegative integer numbers such that 
.
.
Day 2 - 11 May 2014
2 It is given regular 
-sided polygon, 
. 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?
-sided polygon, . 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?
3 Let and be foots of altitudes from and of triangle 
, be intersection point of angle bisector from with side , and 
, 
and 
be circumcenter, center of inscribed circle and orthocenter of triangle , respectively. If 
, prove that 
.
and be foots of altitudes from and of triangle , be intersection point of angle bisector from with side , and , and be circumcenter, center of inscribed circle and orthocenter of triangle , respectively. If , prove that .
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