that are solutions of the equation 
.
be the positive number satisfied 
Find the minimum of 
,
,
.
cube is given, and a 
grid of square unit cells is drawn on each of its six faces. A beam is a 
rectangular prism. Several beams are placed inside the cube subject to the following conditions:
faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 
possible positions for a beam.)
faces of each beam touch either a face of the cube or the interior of the face of another beam.
have the property that there exists an infinite sequence 
of nonzero integers such that the equality ![\[a_k+2a_{2k}+\ldots+na_{nk}=0\]](http://latex.artofproblemsolving.com/5/b/a/5ba1051daa520af4e7a53ece725c400b229075cf.png)
holds for every positive integer 
.
be a given point inside quadrilateral 
.
and 
are located within ![\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\]](http://latex.artofproblemsolving.com/d/b/3/db31d11ec24322724c0e0f1485aadf8864eabf7a.png)
Prove that 
if and only if 
.
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount 
Y by Adventure10
In a certain triangle 
, there are points 
and 
on sides 
and 
, respectively, such that if 
is the intersection of 
and 
, then both 
and 
are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle .
, there are points 
and 
on sides 
and 
, respectively, such that if 
is the intersection of 
and 
, then both 
and 
are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle .
be the intersection point of 
and 
;
and 
are the 
as inscribed angles. Hence 
is cyclic; similarly we prove that 
also is cyclic. We claim that 
and 
,
;
,
.
,
,
is an altitude. Similarly we prove that 
and 
are also altitudes.
as inscribed angles. Moreover, 
,
.
,
.
angles 
)
.
,
,
,
,
are the midpoints of 
,
,
,
and 
are congruent, so 
,
and 
we have 
,
,
,
,
.
cyclic means 
means 
,
as external angle in 
,
becomes 
,
