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and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
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 
, then![\[
2^{(a - 1)/2} \equiv 1 \pmod{a}.
\]](http://latex.artofproblemsolving.com/2/d/c/2dc6cc1fe27cc72e72d63b13cf07939e19051245.png)
, take 
the midpoint of its arc containing 
, draw the circle 
., draw a halfline 
, directed to the halfplane not containing 
, such as 
, it will intersect the second circle at 
; see that for many on the first circle will find a on the second one, and the required angle is not constant.
IS indeed the circumcenter of 
even with the additional information! 
to a point 
such that 
and as 
. So, 
is a cyclic quadrilateral. So, 
. So, 
. Similarly 
. So, 
. So,
.
, which is not there in the question, mods kindly add it.
at is half of that subtended by them at some other point , it does NOT imply that is the circumcentre of 
.
and 
.
be a convex quadrilateral in which 
a
.
is the point of intersection of 
and 
,
.
,
