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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.
consecutive even numbers, such that the square of their product divides the sum of their squares.
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
be a function differentiable at 0 with 
. Find
, 
, 
, 
.
and its absolute and relative error, known that its absolute error is equal or lower than 
If G has a normal subgroup of order 
then prove that 
is abelian without using Sylow Theorems.
![\[
\int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx
\]](http://latex.artofproblemsolving.com/e/a/9/ea9f3a093b1d4c0fa39e5b6d42afa2f71dace1e0.png)
a prime number, with 
and 
, vector space.
is a group?
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
be a continuous function such that 
for all 
. Show that 
for 
.
, 
, 
is the point on 
such that 
. Let 
be the circle through tangent to 
at , and 
the circle through tangent to at 
. Let 
be the second intersection of and . Prove that lies on .
be a point on ray beyond is tangent to circle 
and is tangent to circle 
is cyclic.
, giving the required collinearity.
be the intersection of and . We will prove that 
, which implies![\[
(!) \quad AB \text{ is tangent to } (CF'B).
\]](http://latex.artofproblemsolving.com/0/9/1/0912a5d89c9b96991df593e9811fe2ccb3ad79db.png)
Due to the Power of a Point theorem,![\[
AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/c/7/c/c7ce7f6916003527f4d1812c1cfcaf61d3bf0bf6.png)
Since 
, we have![\[
AB^2 = AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/e/f/f/eff6902bf20790e7dfdb273eee89ca8dac4c58c8.png)
. Note that 
is the radical axis of 
, and so this implies that 
lies on extended, which then implies lies on .
be 
,
is tangent to 
and 
meet on