[/quote]
,
be positive integers and let 
be a non-increasing list of positive real numbers. Prove that there exists 
sets 
which partition the set 
such that 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
,
,
,
.
and 
are marked, draw the line connecting 
and 
are drawn, mark the intersection of 
is drawn, draw a line parallel to 
using these operations.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
,
to be the set of all triplets 
of natural numbers, with 
odd and 
,
and 
.
be positive real numbers . Prove that
and 
in the plane, draws the perpendicular bisector of 
.
.
be the power of 
be the power of 
.
satisfy the equality
Find the maximum possible value of 
.
board. Julia
,
be its 
be the projection of 
.
be the intersection of 
and 
be the points on 
,
are on a line perpendicular to 
intersect 
,
intersects the circumcircle of 
.
Y by Adventure10, Mango247
In the space 
points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
connected by the smallest distance are connected to each other, so we have at most 
edges in our figure (because the edge going out from 
.
