[/quote]
,
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)
such that 
for all 
?
be a regular heptagon with its center 
.
is the orthocenter of triangle 
,
is the incenter of triangle 
.
be the midpoint of 
and 
be the intersection point of 
and 
.
is the circumcenter of triangle 
.
concur at a single point.
be a cyclic quadrilateral such that 
is a diameter of it's circumcircle. Suppose that 
intersect at 
and 
at 
,
and 
at 
,
be a point on 
is perpendicular to 
if and only if 
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
,
and 
intersect at points 
and 
.
and 
intersect in two points 
and 
.
and 
are concurrent.
is a natural number.
such that 
.
be an increasing sequence of positive integer such that 
.
there exists 
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 
.
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
are both on 
and 
,
,
and 
intersects on 
and 
Prove that
Let 
and 
Prove that
.
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)
Y by
In Mexico, there live 
Mexicans, some of whom know each other. They decided to play a game. On the first day, each Mexican wrote a non-negative integer on their forehead. On each following day, they changed their number according to the following rule: On day 
, each Mexican writes on their forehead the smallest non-negative integer that did not appear on the forehead of any of their acquaintances on day 
. It is known that on some day every Mexican wrote the same number as on the previous day, after which they decided to stop the game. Determine the maximum number of days this game could have lasted.
Proposed by Pavle Martinović
Mexicans, some of whom know each other. They decided to play a game. On the first day, each Mexican wrote a non-negative integer on their forehead. On each following day, they changed their number according to the following rule: On day 
, each Mexican writes on their forehead the smallest non-negative integer that did not appear on the forehead of any of their acquaintances on day 
. It is known that on some day every Mexican wrote the same number as on the previous day, after which they decided to stop the game. Determine the maximum number of days this game could have lasted.Proposed by Pavle Martinović
