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.
with 
, let 
denote the incenter and 
the midpoint of side 
. The line through 
perpendicular to 
intersects the tangent from to the incircle (different from line ) at a point 
> Show that is tangent to the circumcircle of triangle 
.![$P \in \mathbb R[x,y]$](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.
, such that 
and 
are both integers.
be real numbers such that 
Prove that 
Let be real numbers such that 
Prove that 
. I couldn't solve this and was wondering if any of you can find the pattern
. I couldn't solve this and was wondering if any of you can find the pattern
.
, and thus the next term is 
Prove that
and 
Prove that
,
![\[
1, 2, 5, 40, 1280, ?
\]](http://latex.artofproblemsolving.com/a/b/d/abdadcc9d1511e0bc50630d0764932c6723f6123.png)
![\[
a_n = \frac{381}{8}n^{4} - \frac{1885}{4}n^{3} + \frac{13103}{8}n^{2} - \frac{9313}{4}n + 1115
\]](http://latex.artofproblemsolving.com/3/a/5/3a5bf7857ee63dfefc85efdd2f73062889f38dd4.png)
:![\[
a_1 = \frac{381}{8}(1)^4 - \frac{1885}{4}(1)^3 + \frac{13103}{8}(1)^2 - \frac{9313}{4}(1) + 1115 = 1
\]](http://latex.artofproblemsolving.com/a/4/b/a4b8690544780a4953ff30b7e9bfd650d5c05c64.png)
:![\[
a_2 = \frac{381}{8}(2)^4 - \frac{1885}{4}(2)^3 + \frac{13103}{8}(2)^2 - \frac{9313}{4}(2) + 1115 = 2
\]](http://latex.artofproblemsolving.com/4/0/1/401ac0e6b9e04a3bbb18b93adf3afd46aa401c2a.png)
:![\[
a_3 = \frac{381}{8}(3)^4 - \frac{1885}{4}(3)^3 + \frac{13103}{8}(3)^2 - \frac{9313}{4}(3) + 1115 = 5
\]](http://latex.artofproblemsolving.com/7/f/c/7fc785dbf6d9a0992d3bb5e2011bc4db67ef1c02.png)
:![\[
a_4 = \frac{381}{8}(4)^4 - \frac{1885}{4}(4)^3 + \frac{13103}{8}(4)^2 - \frac{9313}{4}(4) + 1115 = 40
\]](http://latex.artofproblemsolving.com/2/1/b/21b4e2e4d19b6d5c983417f70abf485ad3eb155a.png)
:![\[
a_5 = \frac{381}{8}(5)^4 - \frac{1885}{4}(5)^3 + \frac{13103}{8}(5)^2 - \frac{9313}{4}(5) + 1115 = 1280
\]](http://latex.artofproblemsolving.com/b/d/9/bd9f2528ec68ad7182630c86181261f2eeabad17.png)
:![\[
a_6 = \frac{381}{8}(6)^4 - \frac{1885}{4}(6)^3 + \frac{13103}{8}(6)^2 - \frac{9313}{4}(6) + 1115
\]](http://latex.artofproblemsolving.com/4/a/8/4a8c57a4df9d6246743b76544bd06d424c99c27f.png)
![\[
a_6 = \frac{381}{8} \times 1296 - \frac{1885}{4} \times 216 + \frac{13103}{8} \times 36 - \frac{9313}{4} \times 6 + 1115
\]](http://latex.artofproblemsolving.com/b/a/d/bade9652d9835650aa5c7ae840e9564696946715.png)
![\[
a_6 = 69420
\]](http://latex.artofproblemsolving.com/a/9/c/a9cc1cb8407e92daee7dff378e21ddbde47fd9fd.png)
![\[
\boxed{69420}
\]](http://latex.artofproblemsolving.com/2/f/1/2f16609b47f00089d42077ac2f80a849752c8766.png)
.
