[/quote]
,
and 
Prove that
Let 
Prove that
be a positive integer. We start with 
and 
if 
then find the coefficient of 
in 
in terms of 
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
and 
Prove that
be an acute triangle and let 
be the midpoint of 
.
passing through 
and 
respectively. Let 
is a parallelogram. Suppose that 
.
such that![\[f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)\]](http://latex.artofproblemsolving.com/9/8/a/98ab64d17bc7d6bf7533b0874177fc963405c1d8.png)
.
satisfying the following property: if 
,
and 
are integers such that 
,
such that 
.
be functions 
such that for all reals 
,
Prove that either 
is the identity function or 
is periodic.
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
be the set of real numbers. Determine all functions ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
![\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]](http://latex.artofproblemsolving.com/e/b/4/eb4e0d6580d9d3d6a627e39837edb3d1943d4596.png)
Y by Adventure10, Mango247
A circle that passes through the vertex 
of a rectangle 
intersects the side 
at a second point 
different from 
A line passing through 
is tangent to this circle at a point 
and the circle with center and passing through 
intersects the side 
at the point 
Show that if 
then 
of a rectangle 
intersects the side 
at a second point 
different from 
A line passing through 
is tangent to this circle at a point 
and the circle with center and passing through 
intersects the side 
at the point 
Show that if 
then 
is the midpoint of 
,
.
.
.
.
.
and 
are complementary, so 
.
and 
to each other. Let 
and 
.
and 
,
.
.
,
.
.
gives 
is tangent to 
.
is cyclic. Hence, 
.
