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, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from 
to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle. arcs which Bunbun traveled, over all possible labellings of the points.
and 
be points on a plane such that 
, where is a positive integer. Let 
be the set of all points such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
be reals such that 
. Prove that
Let be reals such that 
. Prove that
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
. Find the minimum and maximum values of 
+ 
+ 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Prove that 
be real numbers such that 
Prove that
Find 
in the interior of a convex quadrilateral 
to the sides 
is a constant, prove that 
