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.
. Denote by 
the number of changes of signs in the
Then prove that
.
be an acute triangle with orthocenter 
, and let 
be a point on the side 
, lying strictly between 
and 
. The points 
and 
are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
greater than 
, such that, if 
, then 
such that 
is a perfect square but none of the integers 
are perfect squares.
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
and are externally tangent at a point 
. Let 
be a line tangent to at 
and at 
. Let 
and 
be diameters in and respectively. Suppose points 
and 
lies on such that 
and 
are tangent to , and points 
and 
lies on such that 
and 
are tangent to ., , , lie on a circle 
., , , determine a quadrilateral with an incircle 
, and its radius is 
times the radius of . be a triangle and let 
and 
denote its incentre and circumcentre respectively. Let 
be the circle through and which is tangent to the incircle of the triangle ; the circles 
and 
are defined similarly. The circles and meet at a point 
distinct from ; the points 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
. lies on 
. The Simson line of intersects Line 
at . and are the reflections of and over . Prove that the circles 
and 
intersects at a point on line .
+1 w![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
, 
, 
, 
be non-negative real numbers such that![\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]](http://latex.artofproblemsolving.com/d/e/8/de8ccd816cfe032a97d940982192b1abea21dc18.png)
prove that : ![\[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]](http://latex.artofproblemsolving.com/a/f/c/afc556cd8f5c7f4dedf153ffb92fb1f978c56823.png)
and 
Prove that
and 
Prove that
,
such that 
can be written in decimal notation using only digits 1 and 2.
