[/quote]
,
and 
be touch points of the incenter of 
at 
and 
,
and 
be the circumcenter of triangles 
and 
,
and 
concurrent.
are such that 
,
be a sequence of strictly positive real numbers. For each nonzero positive integer 
,![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
![\( P(x) \in \mathbb{Z}[x] \)](http://latex.artofproblemsolving.com/2/1/5/215e7ef672c0effe96e63ac9254fbd7356e17637.png)
for which there exists an integer 
such that 
for integers 
.
with 
vertices may have, so that after deleting an arbitrary cycle, 
be nonzero vectors in 
-
are real numbers. Show that there exists 
such that![\[\left|\left\langle\sum_{i=1}^t\varepsilon_iv_i,\frac{v_k}{|v_k|}\right\rangle-m_k\right|\ge |v_k|\]](http://latex.artofproblemsolving.com/6/c/6/6c6f82687e0fbf781f18caed5ba6fdbe9bc1125b.png)
holds for all 
be positive real numbers such that 
.
,
and 
intersect at 
intersect at 
and 
intersect at 
intersects line 
at 
,
intersects line 
and 
,
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
,
.
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.
Y by centslordm, HWenslawski, megarnie
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon 
, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at 
, 
, and 
are 
and 
meters, respectively. What is the height, in meters, of the pillar at 
?
, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at 
, 
, and 
are 
and 
meters, respectively. What is the height, in meters, of the pillar at 
?
be the heights at the vertices. Note that 
,
.
,
.
is a line, 
corresponds to the height at the center, and similarly 
and 
do too, therefore they must be equal.
and 
and so the height at the center is 
and 
.
units. Since the side length of the hexagon doesn’t matter, let it be 
.
.
,
-
,
,
is the side length of 
Therefore, the equation of the plane is
Therefore, if 
The answer is 
.
:diablo:
,
.
.
.
,
.
and 
.
,
.![\[A=(0,0,0), B=(1,0,0), C=\left(\frac{3}{2},\frac{\sqrt{3}}{2},0\right), D=(1,\sqrt{3},0), E=(0,\sqrt{3},0), F=\left(-\frac{1}{2},\frac{\sqrt{3}}{2},0\right)\]](http://latex.artofproblemsolving.com/3/a/5/3a5ae90aa6d232815fbac539a6e5e7a78eb831d3.png)
![\[(0,0,12), (1,0,9),\left(\frac{3}{2},\frac{\sqrt{3}}{2},10\right), (1,\sqrt{3},d'), (0,\sqrt{3},e), \left(-\frac{1}{2},\frac{\sqrt{3}}{2},f\right)\]](http://latex.artofproblemsolving.com/0/d/f/0df5333b36567d9324489b0993f764acb249d044.png)
All these points must be on the same plane.
.![\[12c+d=0, a+9c+d=0, \frac{3}{2}a+\frac{\sqrt{3}}{2}b+10c+d=0\]](http://latex.artofproblemsolving.com/c/1/a/c1abefcddd13f019d45163c05e744eeba49f1c55.png)
.![\[14.5c+d+\frac{\sqrt{3}}{2}b=0\implies \frac{\sqrt{3}}{2}b=-\frac{5}{2}c\implies \sqrt{3}b=-5c.\]](http://latex.artofproblemsolving.com/8/f/2/8f2ffd339137d89078a2c1607d9293b894cf01a1.png)
.
is 
Also, linearity gives 
since 
are collinear. Thus 
,
,
.
.
,
is the desired height. Note that 
must be coplanar.
.
.
by plugging into point 
by plugging into point 
.
.
,
instead of 
