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.
touch sides 
and 
at 
and 
respectively. Let 
and 
denote the circumcircles of triangle 
and 
respectively.
and 
intersect at 
and 
and 
intersect at 
and 
and intersect at 
and 
Prove that 
and intersect in a common point.
Show that 
and 
are concurrent.
such that, for all positive integers 
and 
, if divides 
, then divides 
.
+1 w
be a trapezoid inscribed in circle 
, 
. Let 
is the symmetric point of across 
, 
intersects at 
. Let 
is midpoint of and 
is orthocenter of triangle 
. On 
take a point 
so that 
. Prove that: 
coincide
, let 
. Point 
lies on segment 
such that 
. Let 
and 
be the circumcenter and incenter of triangle 
, respectively. Let 
and 
be the circumcenter and incenter of triangle 
, respectively. Suppose that lines 
and 
intersect at point 
. Find the length of 
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
, point on 
satisfies 
and 
. be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds. is an isosceles triangle.
be a positive integer divisible by 3, and let 
be non-negative real numbers satisfying 
. Define![\[
M_n = \min_{1 \le i \le n} \{x_i x_{3i}\},
\]](http://latex.artofproblemsolving.com/d/b/b/dbbf0f6bf20f506d6cb63357b06ec6c52bebed10.png)
where the indices are taken modulo . Determine the maximum possible value of 
.
with integer coefficients which satisfy the following conditions:
is a positive integer for any positive integer 
.
for all positive integers .
+1 w
be the set of positive real numbers. Find all functions 
such that, for all 
,
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
