[/quote]
,
be inscribed in the circle 
.
intersects 
and 
at points 
and 
,
be the reflection of 
be the reflection of 
of triangle 
lies on the circle 
and 
coincide.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
and 
.
?
lies inside triangle 
and 
.
.
.
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
be different real numbers. Prove that![\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]](http://latex.artofproblemsolving.com/6/5/2/6525fc6b451ff5a3c83ff2a0541b063a7cf065c3.png)
satisfying
for all real numbers 
and 
.
be its altitudes. Let 
and 
be the circumcenters of triangles 
and 
.
be the second intersection of the circumcircles of triangles 
and 
.
,
,
-
are concurrent.
that satisfy the condition 
for the natural number 
Here 
and 
are natural numbers. Is there a natural number 
such that each of the numbers 
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
we have 
.
.
Y by
1. There are 
circles 
, which are externally touch each other consecutively in a cycle, i.e., pairs of touching- circles: 
and 
and 
and 
and . Prove that if the centers of the circles form a convex quadrilateral, then it is tangential.
2. Lines
and 
touch four circles described in the previous paragraph. Line separates circles and 
from 
and 
, i.e. circles and are on the same side of the straight line , and and - on the other. Similarly, the line separates and from and . Prove that the quadrilateral formed by the centers of the circles is a rhombus.
3. There are
spheres that touch the "dice": 
and (consecutively in a cycle) and 
and 
(also consecutively in a cycle), as well as and 
and 
and 
and . There are also three planes, each touching all eight spheres, where the first separates the spheres 
from the spheres 
, the second - spheres 
and 
from 
and , and the third - 
and from 
and 
. Prove that the centers of the spheres and lie in the same plane.
4. The centers of the eight spheres described in the previous paragraph are the vertices of a hexagon with quadrangular faces. Prove that if this hexagon is inscribed, then it is a cube.
circles 
, which are externally touch each other consecutively in a cycle, i.e., pairs of touching- circles: 
and 
and 
and 
and . Prove that if the centers of the circles form a convex quadrilateral, then it is tangential.2. Lines
and 
touch four circles described in the previous paragraph. Line separates circles and 
from 
and 
, i.e. circles and are on the same side of the straight line , and and - on the other. Similarly, the line separates and from and . Prove that the quadrilateral formed by the centers of the circles is a rhombus.3. There are
spheres that touch the "dice": 
and (consecutively in a cycle) and 
and 
(also consecutively in a cycle), as well as and 
and 
and 
and . There are also three planes, each touching all eight spheres, where the first separates the spheres 
from the spheres 
, the second - spheres 
and 
from 
and , and the third - 
and from 
and 
. Prove that the centers of the spheres and lie in the same plane.4. The centers of the eight spheres described in the previous paragraph are the vertices of a hexagon with quadrangular faces. Prove that if this hexagon is inscribed, then it is a cube.
