of integers satisfying![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](http://latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
.
.
,
and 
and so forth.
.
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
,
,
,![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
be points on a plane such that 
,
is a positive integer. Let 
be the set of all points 
,
is a real number. The path that 
be reals such that 
. Prove that
Let 
. Prove that
and 
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 Adventure10
Let 
be an acute-angled triangle with 
, and let points 
and 
be chosen on the side 
and 
respectively in such a way that 
. The circumcircle of 
intersects the line at 
and 
and the line 
at and 
. Prove that is the circumcentre of 
.
be an acute-angled triangle with 
, and let points 
and 
be chosen on the side 
and 
respectively in such a way that 
. The circumcircle of 
intersects the line at 
and 
and the line 
at and 
. Prove that is the circumcentre of 
.
is cyclic and 
,
and 
are similar. Since 
,
.
,
and hence 
.
,
.
