[/quote]
,
be positive real numbers such that : 
. Prove that : 
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
be a set with seven elements, where each element is a positive integer not exceeding 
.
)![\[
x_1 + x_2 + \cdots + x_t = x_{t+1} + x_{t+2} + \cdots + x_m
\]](http://latex.artofproblemsolving.com/7/0/7/707e7e9a0ce45adb307545bbc88c1f0bf90ca0c9.png)
has a solution in the set 
,
are all distinct.
such that: 
.
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then 
touch sides 
at 
,
lie on the line through 
perpendicular to 
.
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
and 
.
bisects segment 
.
can we find three positive integers 
,
and 
such that 
?
satisfy both of the following conditions.
,
,
.
that satisfies the equation 
and 
.
,
,
-
,
.
at 
.
parallel to 
,
meets 
.
is constant.
.
be the feet of the altitudes from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
. Prove that
Where 
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After a long while, I'm back again.
But this time, with a really huge collection of problems...
I didn't want to share this since it was the source of the book I'm writing, but now I think this is not a good reason...
Dedicated to all AoPS users.
PS. This was a boring work of mine for about one week. All the problems have the links in the forum and I have mentioned them all in the PDF file. I edited the first 600 problems in order to fix the LaTeX/Math errors of this file. Please, if you can help me edit this huge collection, send me a PM.
But this time, with a really huge collection of problems...
I didn't want to share this since it was the source of the book I'm writing, but now I think this is not a good reason...
Dedicated to all AoPS users.
PS. This was a boring work of mine for about one week. All the problems have the links in the forum and I have mentioned them all in the PDF file. I edited the first 600 problems in order to fix the LaTeX/Math errors of this file. Please, if you can help me edit this huge collection, send me a PM.
i
Thank you.
and 
are positive integers satisfying 
.
.
