[/quote]
,
be reals such that 
and 
.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
be positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be real numbers such that 
.
Let 
.
and 
,
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
and 
and 
.
,
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
and 
points on a plane is 
,
points of a plane, no three of which are collinear, consisting of 
,
be the set of all positive integers not exceeding ![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
Y by Fedor Bakharev, Adventure10
A triangular pyramid 
is given. A sphere 
is tangent to the face 
and to the planes of other faces in points don't lying on faces. Similarly, sphere 
is tangent to the face 
and to the planes of other faces in points don't lying on faces. Let 
be the point where is tangent to , and let 
be the point where is tangent to . The points 
and 
are chosen on the prolongations of 
and 
over and such that 
and 
. Prove that the distances from the points , to the midpoint of 
are the same.
is given. A sphere 
is tangent to the face 
and to the planes of other faces in points don't lying on faces. Similarly, sphere 
is tangent to the face 
and to the planes of other faces in points don't lying on faces. Let 
be the point where is tangent to , and let 
be the point where is tangent to . The points 
and 
are chosen on the prolongations of 
and 
over and such that 
and 
. Prove that the distances from the points , to the midpoint of 
are the same.This post has been edited 1 time. Last edited by XbenX, May 1, 2019, 2:38 PM
of tetrahedron is tangent to face 
and define 
analogously. Let 
and 
By homothety it follows 
are cyclic quadrilaterals.
As well-known 
are isogonal conjugates wrt 
therefore 
be a rotation wrt 
which maps 
onto plane 
to the same semiplane wrt 
Clearly 
and 
By claim points 
lie on 
so they are symmetric wrt perpendicular bisector of 
This completes proof.
