[/quote]
,
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
and circumcenter 
.
,
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its circumradius. Let 
,
across the line 
,
across the line 
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
.
is a prime number
is inscribed in circle 
.
its 
.
meets 
at 
meets 
a second time at 
meets 
a second time at 
.
passes through the antipodal to 
intersects the lines 
,
at points 
,
,
of the segments 
,
,
,
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
Y by
Recall that the similarity of a figure with a coefficient 
is a transformation such that any two points 
and 
of the figure are associated with points 
and 
such that 
. A figure 
is called similar to a figure 
with coefficient 
if there is a similarity with coefficient that transforms into .
1. The right-angled triangle
is inscribed in a similar triangle 
, the names of the vertices are corresponding, while the vertex 
lies on 
on 
, and 
on 
. Find all possible values for the coefficient of similarity.
2. Triangle
is inscribed in triangle , similar to it with coefficient 
, the names of the vertices correspond responsible. In this case, vertex lies on on 
on , 
, 
. Prove that 
.
3. The heights of the tetrahedron intersect at one point, and the tetrahedron with vertices at the bases of the heights is similar to the original one. Prove that the original tetrahedron is regular.
4. The tetrahedron with vertices at the bases of heights drawn from the vertices of the original tetrahedron turned out to be correspondingly similar to the initial one. Prove that the square of the similarity coefficient is
, where 
is the dihedral angle at some edge, and 
is the angle between this edge and the opposite one, as between crossing lines.
is a transformation such that any two points 
and 
of the figure are associated with points 
and 
such that 
. A figure 
is called similar to a figure 
with coefficient 
if there is a similarity with coefficient that transforms into .1. The right-angled triangle
is inscribed in a similar triangle 
, the names of the vertices are corresponding, while the vertex 
lies on 
on 
, and 
on 
. Find all possible values for the coefficient of similarity.2. Triangle
is inscribed in triangle , similar to it with coefficient 
, the names of the vertices correspond responsible. In this case, vertex lies on on 
on , 
, 
. Prove that 
.3. The heights of the tetrahedron intersect at one point, and the tetrahedron with vertices at the bases of the heights is similar to the original one. Prove that the original tetrahedron is regular.
4. The tetrahedron with vertices at the bases of heights drawn from the vertices of the original tetrahedron turned out to be correspondingly similar to the initial one. Prove that the square of the similarity coefficient is
, where 
is the dihedral angle at some edge, and 
is the angle between this edge and the opposite one, as between crossing lines.This post has been edited 4 times. Last edited by parmenides51, Jun 21, 2022, 2:22 AM
