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.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a triangle and let be its circumcenter and its incenter. be the radical center of its three mixtilinears and let be the isogonal conjugate of .
be the Gergonne point of the triangle .
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
we have 
.
.
,
,
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)
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 
.
where the 
are positive integers such that each 
and none of two consecutive terms when summed are equal to 
.
as the number of sequence 
such that 
. Define also Conclude first the recurence 
.
