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.
,
,
positive real numbers. Prove that:![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]](http://latex.artofproblemsolving.com/f/b/5/fb5a179f427615c8bc2da46164ecd9a3484d6625.png)
we have 
.
.
such that![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](http://latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer 
.
be an integer. Prove that there exists a positive integer with the following property: For each positive integer , there is a prime number 
(possibly depending on 
) such that 
is divisible by , but not divisible by 
.
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
, find the locus of all points 
such that 
(i.e: sum of powers of point with respect to the two circles is zero).
![\[
\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),
\]](http://latex.artofproblemsolving.com/4/c/b/4cba741142419b031efdeba713748762fa6b2aac.png)
where 
runs over the set of permutations of 
.
, let 
denote the integers modulo , and let ![$\mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/e/5/a/e5a21492095c9116b100e2f7626a70cc26834e82.png)
be the set of polynomials with coefficients in . Find all for which there exists a quartic polynomial ![$P(x) \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/7/a/a/7aa574df0669ce78536b1ecc5c949120aa2b3009.png)
such that for all integers 
, there exists some integer 
such that 
. (Note that there are 
quartic polynomials in in total.)![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
, and 
Determine 
.
has at least one pair of positive integer solution 
for each positive integer 
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
, by the midpoint of , and by 
the orthocenter of . Let 
be the point of intersection of the circumcircle 
of the triangle and the half line 
, and 
be the point of intersection (other than 
) of the line 
and the circle . Prove that 
must hold.
the length of the line segment .)
two not intersecting circles 
and 
are drawn such that is tangent to 
and 
at points and 
respectively, and is tangent to 
and 
at 
and 
respectively. It is known that 
. and .
be an acute-angled triangle with 
and let 
be the foot of the
-angle bisector on . The reflections of lines and 
in line meet and at points and respectively. A line through meets and at and 
respectively such that and 
while lies strictly between 
and . Prove that the circumcircles of
and 
are tangent to each other.
be a cube with an edge of 
m. Determine the minimum value of the length the segment with the ends on 
and 
and forms an angle of 
with the plane of the face .
