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.
that has 8097 distinct points with area of a triangle that has 3 points belong to all 
. Prove that there exists a triangle 
that has the area 
contains at least 2025 points that belong to ( each of that 2025 points can be inside the triangle or lie on the edge of triangle )X
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses. for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
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.
and 
Prove that
and 
Prove that
