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 a nonempty set of positive integers such that:
if 
then 
. for any prime 
, there exists 
such that 
. is finite.
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
+1 w
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](http://latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
such that 
for all 
and 
and coefficients in the set 
. Prove that the probability for this polynomial to be special is between 
and 
, where a polynomial 
is called special if for every 
in the sequence 
there are infinitely many numbers relatively prime with 
.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
contestant of EGMO are named 
. After the competition, they queue in front of the restaurant according to the following rules.
with 
.
has at least other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly positions. has fewer than other contestants in front of her, the restaurant opens and process ends. the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
numbers 
and 
numbers 
are written in some order.
.
, where 
is a prime number. Show that if is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
reals, and 
.
?
be a triangle with 
. Let be the orthocenter of . Point is the midpoint of and point lies on the side such that 
. Prove that
.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
is the set of nonnegative integers
is not always true. Check 
.
. Prove that
and 
. Prove that
Thank lbh_qys.
and 
?
and 
.
for all positive integers 
,
,![\[f(x+1)\ge f(x)+N\]](http://latex.artofproblemsolving.com/e/6/d/e6d7a6bcac7cb561489adbf005fd4c784afac66d.png)
so 
is infinitely large, contradiction.