AoPS Academy
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.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
be the roots of the equation 
. What is the value of the expression
?
such that
Proposed by Ngo Van Trang, Vietnam
such that 
. Find the value of 
and 
.
and 
such that 
is 
, then find the value of
.
be the set of integers. Determine all functions 
such that for all integer :
of a triangle 
are positive integers. Let
for any positive integer .
and 
, determine all possible perimeters of the triangle .
has two roots, where one is twice of the other, find all possible values of .
Find the value of 
such that 
is an integer.
is 
in triangle , find the ratio of the side length 
to the side length 
.
.What is the least possible positive sum of the three integers?
, points 
, 
, and lie on sides 
, 
, and , respectively, such that
, 
, 
, 
, 
, 
.
?
where 
is a parallelogram.
.
is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
, point 
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
, to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
.
.
.
and 
be the two foci of ellipse 
. is a point on . Let 
be the center of the excircle of 
. When 
, find the minimum eccentricity of .
Prove that
Let 
Prove that
as 
Prove that 
is prime for all 
Prove that
