AoPS Academy
be an acute-angled triangle in which 
and 
. Let point 
lie on segment 
and point 
lie on segment 
such that 
, 
and 
. Let 
be the circumcenter of triangle 
, 
the orthocenter of triangle , and 
the point of intersection of the lines 
and 
. Prove that , , and are collinear.
, there are exactly two or three positive real numbers 
satisfying the equation 
.
. Find the minimum value of 
with two altitudes 
and 
.Let 
be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
and 
be positive integers. Prove that![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](http://latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
is not a perfect square.
matrices
matrices matrices
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
is never a square of an integer.
are complex numbers with absolute value 
. For 
define 
. Given that the sequence 
converges. Show that,
.
be a positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer 
such that 
.
is increasing , because 
is increasing and unbounded, and 
so take all non-negative values is an arbitrary irrational number” xD. I was confused.
![\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}}
\, \mathrm{d}u
\]](http://latex.artofproblemsolving.com/9/1/1/91165e6b531233c56b799ed32ded982d57b6c87a.png)
is dense in 
so we split the interval into