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
In a parallelogram 
, 
is the midpoint of 
and 
an arbitrary point on the side 
. Let 
be the intersection of 
and 
, and 
the intersection of 
and 
. Prove that the triangles 
and 
have equal areas.
, 
is the midpoint of 
and 
an arbitrary point on the side 
. Let 
be the intersection of 
and 
, and 
the intersection of 
and 
. Prove that the triangles 
and 
have equal areas.
