[/quote]
,
be non-negative real numbers such that 
Prove that
over 
.
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
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 
.
be a scalene triangle with orthocenter 
and circumcenter 
.
,
,
.
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 
Y by Adventure10
Let 
be two circles of equal radius, disjoint, of centers 
, such that 
is to the left of 
. Let 
be a line parallel to the line 
, secant to both circles. Let 
be a point of , to the left of and 
a point of , to the right of such that the tangents of to and of a form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.
be two circles of equal radius, disjoint, of centers 
, such that 
is to the left of 
. Let 
be a line parallel to the line 
, secant to both circles. Let 
be a point of , to the left of and 
a point of , to the right of such that the tangents of to and of a form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.This post has been edited 1 time. Last edited by parmenides51, May 30, 2019, 9:02 PM
Reason: source edit
Reason: source edit
and 
intersect at 
.
and are tangent to 
be 
and 
,
and are tangent to 
be 
and 
.
and 
and the radius of this circle be 
.
Similarly, since 
and 
and the radius of this circle be 
.
And finally, since 
,
.
.
Hence, 
is the circle that's tangent to the four sides of the quadrilateral.
