[/quote]
,
such that 
.
such that for all 
.
meets 
at points 
,
to 
is 
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
are chosen such that when going around the triangle, the points occur in the order 
.![\[
AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2.
\]](http://latex.artofproblemsolving.com/4/c/f/4cf9e7bc009736836bcd10c1d8ba46bd3a91cd7b.png)
Prove that the perimeters of the triangles formed by the triplets 
and 
are equal.
and 
,
such that![\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d20546de5736770a9929ac86163e3b06553b7d.png)
![\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/6/6/8/668d53723d307556536736fda0e4a19be848834f.png)
![\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/1/9/f/19f2627a432354a757d3c9e8ba42e96f83f46726.png)
So:![\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/0/7/3/073602693604bbf3cb1f5c1c2e5051e8994a5e78.png)
![\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/1/1/4/11441fab3ba768fee7a0eef4dbf68124d6637aec.png)
![\[
(n + a)(n + b) = k^2.
\]](http://latex.artofproblemsolving.com/0/e/c/0ec5e7bac2aebe9ec507727ed21f7ca028655222.png)
Assume 
.![\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]](http://latex.artofproblemsolving.com/7/5/f/75fc52b6662df3a4c0cb2f40f49c4b3351c90a4d.png)
or it could also be that 
.![\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]](http://latex.artofproblemsolving.com/1/f/5/1f5936241f7bebcf15c68fed5267bbfad2d40c2b.png)
![\[
k_1 k_2 = \frac{n + b}{n + a}.
\]](http://latex.artofproblemsolving.com/d/4/c/d4c1958e3554b58c3def282070f849842cc10912.png)
,![\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]](http://latex.artofproblemsolving.com/3/0/6/306a6a00c44d4e4db1b36c7e588f684f28fdbf44.png)
,
must be an integer, which is not possible.
and 
.
be reals such that 
and 
Prove that
Let 
and ![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](http://latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](http://latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
.
.
.
Y by SHARKYKESA, Mango247
Let 
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and 
be the circumcircle of 
. Let 
be a point on . Suppose that the circle centered at passing through 
intersects at distinct points 
and 
. Let 
and 
lie on 
such that 
and 
are tangent to , and denote by 
the intersection of and .
Prove that regardless of the choice of, the triangle 
has constant perimeter.
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and 
be the circumcircle of 
. Let 
be a point on . Suppose that the circle centered at passing through 
intersects at distinct points 
and 
. Let 
and 
lie on 
such that 
and 
are tangent to , and denote by 
the intersection of and .Prove that regardless of the choice of
, the triangle 
has constant perimeter.
