AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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)
such that 
for all 
?
be a regular heptagon with its center 
. 
is the orthocenter of triangle 
, 
is the incenter of triangle 
. Let 
be the midpoint of 
and 
be the intersection point of 
and 
. Assume 
is the circumcenter of triangle 
. Prove that 
concur at a single point.
be a cyclic quadrilateral such that 
is a diameter of it's circumcircle. Suppose that and 
intersect at , 
and 
at 
, 
and 
at 
, and let 
be a point on . Show that 
is perpendicular to 
if and only if is the midpoint of .
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
, and circles of diameters 
and 
intersect at points 
and 
. The circles of diameters 
and 
intersect in two points 
and 
. Prove that the lines 
and 
are concurrent.
such that 
is a natural number.
.
such that 
.
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)
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
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
be an increasing sequence of positive integer such that 
.
there exists 
and 
.
