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 the incentre of acute triangle 
with 
. The incircle 
of is tangent to sides 
, and 
at 
and 
, respectively. The line through 
perpendicular to 
meets at 
. Line 
meets again at 
. The circumcircles of triangle 
and 
meet again at 
.
and 
meet on the line through 
perpendicular to 
.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
integers are such that 
is prime and 
. Prove that there exists a positive integer 
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer 
satisfying 
.
is described on a circle , and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
+1 w
define 
be the product of first 
primes. Determine if there exists positive integers 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be reals such that 
and 
Prove that
Let be reals such that and 
Prove that
Let be reals such that and 
Prove that
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer such that: in every set of 
points of a plane, no three of which are collinear, consisting of points of every color there exists a subset.
?
be an isosceles trapezoid with 
. Let 
and 
symmetric to 
wrt 
. Prove that the quadrilateral 
is cyclic.
and 
. Since 
then
and 
thus in triangle 
be an isosceles trapezoid with . Let and symmetric to wrt . Prove that the quadrilateral is cyclic.
mean parallel?
mean parallel?
be an isosceles trapezoid, where 
and is cyclic. Given that 
, we can deduce the following:, it follows that 
. is cyclic, we have 
.
be a point such that 
, then 
.
be the intersection point of 
and 
. Since is symmetric to 
, we have 
and 
.
is isosceles and , it follows that 
.
and , we have that 
is the angle bisector, and thus 
.
, which implies that quadrilateral 
is cyclic.
and 
Prove that
and 
Prove that
