1. For a given chord 
of a circle discussed the triangle 
, whose base is the diameter 
of this circle,which do not intersect the , and the sides 
and 
pass through the ends of 
and 
of the chord . Prove that the heights of all such triangles drawn from the vertex 
to the side , intersect at one point.
2. In a convex
-gon all angles are equal from a certain point, located inside the -gon, all its sides are seen under equal angles. Can we conclude that this -gon is regular?
3. Given positive numbers 
with 
. Prove that 
.
4. Given equation
, with real 
. Prove that 
.
5. Points and are chosen on sides and ,respectively, in a triangle , such that point 
is interserction of lines 
and 
. Given that 
. Prove that 
.
6. a) Among the
pairwise distances between the 
points of the plane, prove that one and the same number occurs not more than 
times.
b) Find a maximum number of times may meet the same number among the
pairwise distances between 
points of the plane.
7. Given that
and 
, for natural . Prove that 
.
8. Given a sequence
of real numbers with 
, where 
. What must be value of 
, so that 
and 
becomes equal?
of a circle discussed the triangle , whose base is the diameter of this circle,which do not intersect the , and the sides and pass through the ends of and of the chord . Prove that the heights of all such triangles drawn from the vertex to the side , intersect at one point. 2. In a convex
-gon all angles are equal from a certain point, located inside the -gon, all its sides are seen under equal angles. Can we conclude that this -gon is regular?
with . Prove that . 4. Given equation
, with real . Prove that .5. Points
and are chosen on sides and ,respectively, in a triangle , such that point is interserction of lines and . Given that . Prove that .6. a) Among the
pairwise distances between the points of the plane, prove that one and the same number occurs not more than times.b) Find a maximum number of times may meet the same number among the
pairwise distances between points of the plane.7. Given that
and , for natural . Prove that . 8. Given a sequence
of real numbers with , where . What must be value of , so that and becomes equal?
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