AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
such that
find the number of ordered real triples 
such that![\[\begin{aligned}
\begin{cases}
x+y^2+z^2=a,\\
x^2+y+z^2=a,\\
x^2+y^2+z=a.
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/9/8/6/986593653bf128bf7b3d497abed78e1459134c5f.png)
such that ![\[ \dfrac{a^2}{2ab^2-b^3+1} \]](http://latex.artofproblemsolving.com/0/d/2/0d2f10786d22af2ea94ebeed6b15e723f4042734.png)
is a positive integer.
?
be a triangle and let 
and 
be variable points on sides 
and 
, respectively, such that 
. Let 
denote the point on 
such that 
is parallel to 
, and let 
denote the point on 
such that 
is parallel to 
. Prove that as 
vary, the circumcircle of 
passes through a fixed point, other than 
.
![\[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\]](http://latex.artofproblemsolving.com/7/9/e/79e388eba20412905964dc2b4ebb491f3c79143b.png)
and the sums ![\[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\]](http://latex.artofproblemsolving.com/3/c/4/3c44766edcc9e5269f0d3396386e41676cbdf42c.png)
For any pair define 
and 
, 
.
, 
such that the sequences 
and 
are unbounded while the sequence 
is bounded?
, 
?
, angle 
. Point 
lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
be a positive integer. Find, in terms of , all polynomials 
with integer coefficients such that for every integer 
, there exists an integer 
such that 
.
.
touch 
at 
, respectively .
touch at 
, respectively .
the midpoint of 
is the Incenter 
of 
.
is the perpendicular bisector of 
, is the image of under homothety 
the radius of is twice the radius of .
be a circumcenter of an acute-angled triangle . Consider two circles 
and 
inscribed in the angle 
in such way that is tangent from the outside to the arc 
of a circle circumscribed about the triangle , and the circle is tangent internally to a circumcircle of triangle . Prove that the radius of is twice the radius .
satisfy that 
.
