AoPS Academy
.
such 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)
be a positive real number. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
,
is midpoint AP,BP. the perpendicular of 
from intersect 
at 
. we'll prove that is circumcenter of 
= 
=
=
then => PKO'H is cyclic quadrilateral => (QED)
be the perpendicular from 
to 
. Let 
,
be the midpoints of 
, . Let 
and 
meet line at 
and 
, respectively. Then since 
and 
, 
and 
are cyclic. Also, 
.
is cyclic, so by the radical axis theorem, 
,, and are collinear, so ,, and are collinear.
, so 
are collinear.
?
. Just note the following:
.
on substitution of values from above. 
. Observe that ![\[\measuredangle APH = \measuredangle CPE = 90^{\circ} - \measuredangle PCE = 90^{\circ} - \measuredangle ABD = \measuredangle APO,\]](http://latex.artofproblemsolving.com/d/c/e/dce88d35f45abbfbc47776f4c1097ad9ad87df97.png)
which is enough to show 
are collinear. 
be an increasing sequence of positive integer such that 
.
there exists 
such that 
and 
of a cyclic quadrilateral 
intersect at 
.
be the circumcenter of triangle 
and 
be the orthocenter of triangle 
.
.
,
,
,
,
a
is a cyclic quadrilateral.
