and 
.
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
and 
Prove that
Let 
and 
Prove that
be natural numbers such that![\[
\frac{a^n - 1}{(a - 1)^n + 1}
\]](http://latex.artofproblemsolving.com/6/2/7/627f1d5f55e7594f2e6dcf5d541e796b7a5bfae2.png)
is a natural number.
is odd.
be a prime divisor of ![\[
a^{(q - 1)/2} \equiv 1 \pmod{q}.
\]](http://latex.artofproblemsolving.com/b/2/1/b2135d8c5526e577722718e6d02bcc52559b0555.png)
3. Prove that if a is prime and 
,![\[
2^{(a - 1)/2} \equiv 1 \pmod{a}.
\]](http://latex.artofproblemsolving.com/2/d/c/2dc6cc1fe27cc72e72d63b13cf07939e19051245.png)
Y by Adventure10, Mango247, and 1 other user
Let 
be a quadrilateral 
. Let 
and 
be two segments perpendicular to the diagonal 
and such that the distance between them is 
, with 
, 
, 
, and 
. Show that the perimeter of hexagon 
does not depend on the position of and so long as the distance between them remains constant.
be a quadrilateral 
. Let 
and 
be two segments perpendicular to the diagonal 
and such that the distance between them is 
, with 
, 
, 
, and 
. Show that the perimeter of hexagon 
does not depend on the position of and so long as the distance between them remains constant.
![[asy]unitsize(4cm);
pair A[] = {
(0,0),
(2,0),
(1,.7)
};
draw(A[0]--A[1]--A[2]--A[0]);
A[3] = .4A[2];
A[4] = (2A[2]+A[1])/3;
A[5] = 4A[2]/3;
A[6] = (A[4].x, A[3].y);
draw(A[3]--(A[3].x,0),red);
draw(A[4]--(A[4].x,0));
draw(A[2]--A[5],dashed);
draw(A[5]--(A[5].x,A[5].y+A[3].y),red);
for(int i=0;i<7;++i) {
label((string) i,A[i]);
}
label("$\alpha$",A[0],2*dir(20));[/asy]](http://latex.artofproblemsolving.com/5/b/5/5b5aefbd7b16fe75c493db5460c2d55c597eba40.png)
.
is invariant.
is invariant.
,
touches point 
)
forces the particular diagram configuration of 
,
is invariant, so 
also is.
is a rhombus, by symmetry we'll be done if we can show that 
is constant, where 
and 
.
and 
be the intersection of diagonals 
and 
.
are constants.
