[/quote]
,
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
,
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
and 
: Prove that :
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
be a positive integer. A positive integer 
is called a benefactor of 
such that 
.
is a benefactor of 
because 
.
benefactors.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
such that for any positive real numbers 
,![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](http://latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
.
meets 
at points 
,
is 
.
intersects line 
at 
,
intersects line 
,
intersect at 
.
is tangent to 
.
denote the number of prime factors of ![\[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]](http://latex.artofproblemsolving.com/2/0/4/2043dca9f2ff2adfd3c8d49fa4dd278871968db0.png)
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
.
.
Y by Adventure10, Mango247
[geogebra]831c935d2f9aa8f009c870e6a3c58466a6ef3cbb[/geogebra]
GIVEN:
A trapezoid
, where 
and 
are right angles.
let
, and 
PROBLEM:
In terms of
and 
, find
(a) the radius
of the inscribed circle
(b) the area of the trapezoid
(c) the perimeter
of the trapezoid
GIVEN:
A trapezoid
, where 
and 
are right angles.let
, and 
PROBLEM:
In terms of
and 
, find(a) the radius
of the inscribed circle(b) the area
of the trapezoid(c) the perimeter
of the trapezoid
be the incenter, 
,
and 
.![\[x^2=(H-r)^2+r^2\\ y^2=(h-r)^2+r^2\\ z^2=(2r)^2+(H-h)^2\\ z^2=x^2+y^2\]](http://latex.artofproblemsolving.com/2/e/6/2e62c216252a5a4c310ab9ef4bb2380e21a777d8.png)
.![\[(H-r)^2+r^2+(h-r)^2+r^2=4r^2+(H-h)^2\]](http://latex.artofproblemsolving.com/f/5/5/f55fffd179f62332420d2d147a7c68e2e8f81b25.png)
![\[r=\frac{Hh}{H+h}.\]](http://latex.artofproblemsolving.com/1/4/6/146ef569129396259855b2438a0475e781f52fa0.png)
![\[A=\frac{2r(H+h)}{2}=Hh.\]](http://latex.artofproblemsolving.com/4/6/4/4649b299ba127f704c85f9b48cfd54ffb84029d5.png)
the perimeter is![\[P=2r+H+h+z=2(H+h).\]](http://latex.artofproblemsolving.com/e/2/6/e2676476385351d26a6343d290f5d12423c3cb63.png)
