Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
,
be the set of all points 
,
is a real number. The path that 
be reals such that 
. Prove that
Let 
. Prove that
be real numbers such that 
Prove that
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
.
+ 
+ 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Find 
Y by Mango247
Let 
be a point inside an acute angle with the vertex 
and 
be the feet of the perpendiculars drawn from onto the sides of the angle. Let point 
belong to the bisector of the angle, 
be the foot of the perpendicular from onto either side of the angle. Denote by 
the midpoints of the segments 
respectively. Known that 
, prove that 
is perpendicular to 
.
Ya. Konstantinovski
be a point inside an acute angle with the vertex 
and 
be the feet of the perpendiculars drawn from onto the sides of the angle. Let point 
belong to the bisector of the angle, 
be the foot of the perpendicular from onto either side of the angle. Denote by 
the midpoints of the segments 
respectively. Known that 
, prove that 
is perpendicular to 
.Ya. Konstantinovski
This post has been edited 1 time. Last edited by parmenides51, Jun 9, 2020, 11:52 PM
